{"id":60,"date":"2017-09-05T08:21:00","date_gmt":"2017-09-05T08:21:00","guid":{"rendered":"https:\/\/www.nickzom.org\/blog\/?p=60"},"modified":"2024-02-05T15:41:56","modified_gmt":"2024-02-05T14:41:56","slug":"how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series","status":"publish","type":"post","link":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/","title":{"rendered":"How To Calculate and Solve Arithmetic Progression In Sequences and Series"},"content":{"rendered":"\n
\n
\n
\n

Arithmetic Progression<\/h2>\n\n\n\n

A sequence is a set of numbers arranged in a definite pattern. Each number is called a term.<\/p>\n\n\n\n

A finite sequence is one that has a last term when listed. For Example: 2,4,6,8,,…,16. An infinite sequence is one that does not have a last term when listed.<\/p>\n\n\n\n

Arithmetic Progression follows the rule of linear sequence which is the sequence in which each term is obtained by adding a distant number (Positive or Negative) to the proceeding terms.<\/p>\n\n\n\n

The constant number is called common difference<\/strong> and it is denoted as “d” while the first term<\/strong> is denoted as “a”.<\/p>\n\n\n\n

If T1<\/sub>, T2<\/sub>, T3<\/sub>, T4<\/sub>, T5<\/sub>, … is a linear sequence, the common difference is obtained as:<\/p>\n\n\n\n\n\n\n\n

T2<\/sub> – T1<\/sub> = T3<\/sub> – T2<\/sub> = T4<\/sub> – T3<\/sub> = T5<\/sub> – T4<\/sub>
Therefore, d<\/b> = Tn<\/sub> – T(n – 1)<\/sub>
where
Tn<\/sub> = nth<\/sup> term of the sequence and T(n – 1)<\/sub> is the (n – 1)th<\/sup< term.<\/sup><\/p>\n\n\n\n

In other words, here is a quick example:
Lets find the common difference in the following arithmetic progression:
a)<\/b> 0,5,10,15,20
b)<\/b> 31<\/sup>\u20444<\/sub>, 51<\/sup>\u20442<\/sub>, 73<\/sup>\u20444<\/sub>, 10, 121<\/sup>\u20444<\/sub><\/p>\n\n\n\n

Solution<\/b><\/h4>\n\n\n\n

a) To find the common difference<\/b> we use the formula, Tn<\/sub> – T(n – 1)<\/sub>
d = T2<\/sub> – T1<\/sub>
T2<\/sub> = 5 and T1<\/sub> = 0
d = 5 – 0
d = 5
Therefore, the common difference for the first series (a) is 5.<\/p>\n\n\n\n

b) Then, by applying the same principle,
d = T2<\/sub> – T1<\/sub>
T2<\/sub> = 51 d = 51<\/sup>\u20442<\/sub> – 31<\/sup>\u20444<\/sub>
d = 21<\/sup>\u20444<\/sub><\/sup><\/p>\n\n\n\n

The nth<\/sup> term of an arithmetic progression is denoted as Tn<\/sub><\/b> or Un<\/sub><\/b>,
if T1<\/sub = a
T2<\/sub> = T1<\/sub> + d = a + d
T3<\/sub> = T2<\/sub> + d = a + d + d = a + 2d<\/sub><\/p>\n\n\n\n

Therefore, this implies that
Tn<\/sub> = a + (n – 1)d
Hence, this formula above is the general formula used in solving arithmetic progression<\/p>\n\n\n\n

Let’s take some examples:<\/h5>\n\n\n\n

a)<\/b> Find the nth<\/sup> term in 34, 40, 46
b)<\/b> The second, third and fourth terms in an arithmetic progression are x – 2, 5 and x + 2 respectively. Therefore, calculate the value of x.
c)<\/b> Find the number of terms in an arithmetic progression given that its first and last terms are x and 31x and its common difference is 5x.
d)<\/b> Find the number of terms and the expression for the nth<\/sup> term of the following arithmetic progression 32, 29, 26, …, -118
e)<\/b> The 8th<\/sup> term of an arithmetic progression is 18 and the 12th<\/sup> term is 26. So, find the first term, common difference and the 20th<\/sup> term.
f)<\/b> The first term of an arithmetic progression is 10, the ratio of the 7th<\/sup> term to the 9th<\/sup> term is 3:5. Then, calculate the common difference of the progression.
g)<\/b> The numbers 11, x, y, 211<\/sup>\u20442<\/sub> from an arithmetic progression. Hence, find the values of x and y.<\/p>\n\n\n\n

Solution<\/b><\/h4>\n\n\n\n

a) We have the formula as Tn<\/sub> = a + (n – 1)d
a = 34
d = T2<\/sub> – T1<\/sub> = 40 – 34 = 6
Therefore, Tn<\/sub> = 34 + (n – 1)6
Tn<\/sub> = 34 + 6n – 6
Tn<\/sub> = 34 – 6 + 6n
i.e. Tn<\/sub> = 28 + 6n<\/p>\n\n\n\n

b)
2nd<\/sup> term = x – 2
3rd<\/sup> term = 5
4th<\/sup> term = x + 2<\/p>\n\n\n\n

Since there is no first term, we would use common difference to calculate the value of x<\/h5>\n\n\n\n

d = T3<\/sub> – T2<\/sub> = T4<\/sub> – T3<\/sub>
d = 5 – (x – 2) = (x + 2) – 5
5 – x + 2 = x + 2 – 5
5 + 2 – 2 + 5 = x + x
10 = 2x
5 = x
Therefore, the value of x is 5.<\/p>\n\n\n\n

c)
a (First Term) = x
Tn<\/sub> = l (Last Term) = 31x
d (Common Difference) = 5x<\/p>\n\n\n\n

Tn<\/sub> = a + (n – 1)d
31x = x + (n – 1)5x
31x = x + 5nx – 5x
i.e. 31x = -4x + 5nx
31x + 4x = 5nx
35x = 5nx
7 = n
Therefore, the number of terms is 7.<\/p>\n\n\n\n

d)
A.P. => 32, 29, 26, …, -118
So, let’s first of all find the number of terms
General Formula: Tn<\/sub> = a + (n – 1)d
a = 32
d = T2<\/sub> – T1<\/sub>
d = 29 – 32 = -3
l = -118<\/p>\n\n\n\n

-118 = 32 + (n – 1)-3
-118 = 32 -3n + 3
So, -118 = 35 – 3n
3n = 35 + 118
3n = 153
n = 51
Therefore, the number of terms is 51.
Then, we find the nth<\/sup> term expression
Tn<\/sub> = 32 + (n – 1)-3
Tn<\/sub> = 32 + 3 – 3n
i.e. Tn<\/sub> = 35 – 3n<\/p>\n\n\n\n

e)
T8<\/sub> = 18, n = 8
T12<\/sub> = 26, n = 12<\/p>\n\n\n\n

First, let’s find the T8<\/sub> and T12<\/sub> expression
T8<\/sub> = a + (8 – 1)d
T8<\/sub> = a + 7d … (I)<\/p>\n\n\n\n

T12<\/sub> = a + (12 – 1)d
T12<\/sub> = a + 11d … (II)<\/p>\n\n\n\n

So, we obtain the values of a and d using simultaneous equation for equation (I) and (II)<\/p>\n<\/div><\/div>\n\n\n\n

<\/p>\n\n\n\n

\n
\"How<\/figure>\n\n\n\n


Using Elimination Method<\/b><\/h4>\n\n\n\n


18 = a + 7d
-26 = a + 11d<\/p>\n\n\n\n

This leaves us with this equation
-8 = -4d
d = -8 \/ -4 = 2
Common Difference is 2<\/p>\n\n\n\n

Using Equation (II) to find the value of a (first term)<\/h5>\n\n\n\n

26 = a + 11d
That is, 26 = a + 11(2)
26 = a + 22
a = 26 – 22
a = 4
First Term is 4<\/p>\n\n\n\n

Ne, we calculate the value of the 20th<\/sup> term, which will be T20<\/sub> = 4 + (20 – 1)2
T20<\/sub> = 4 + 19(2)
T20<\/sub> = 4 + 38
That is, T20<\/sub> = 42<\/p>\n\n\n\n

f)
First Term (a) = 10
T7<\/sub> : T9<\/sub> = 3:5 = 3<\/sup>\u20445<\/sub><\/p>\n\n\n\n


So, by using the general formula: Tn<\/sub> = a + (n – 1)d<\/p>\n\n\n\n

T7<\/sub> = 10 + (7 – 1)d
T7<\/sub> = 10 + 6d<\/p>\n\n\n\n

Then,<\/p>\n\n\n\n

T9<\/sub> = 10 + (9 – 1)d
T9<\/sub> = 10 + 8d<\/p>\n\n\n\n

Since T7<\/sub>:T9<\/sub> = 3:5
It implies that 5T7<\/sub> = 3T9<\/sub><\/p>\n\n\n\n

5(10 + 6d) = 3(10 + 8d)
50 + 30d = 30 + 24d
30d – 24d = 30 – 50
6d = -20
d = -3.3333<\/p>\n\n\n\n

Therefore, the common difference is -3.3333<\/p>\n\n\n\n

g)
A.P. => 11, x, y, 211<\/sup>\u20442<\/sub>
General Formula:<\/b> Tn<\/sub> = a + (n – 1)d
T4<\/sub> = 211<\/sup>\u20442<\/sub>, n = 4, a = 11<\/p>\n\n\n\n

211<\/sup>\u20442<\/sub> = 11 + (4 – 1)d
211<\/sup>\u20442<\/sub> = 11 + 3d
43<\/sup>\u20442<\/sub> = 11 + 3d
Multiplying throughout by 2
43 = 22 + 6d
43 – 22 = 6d
21 = 6d
d = 21\/6
d = 3.5
Then, Let’s find x<\/p>\n\n\n\n

T2<\/sub> = x = 11 + (2 – 1)3.5
x = 11 + (1)3.5
x = 11 + 3.5
That is, x = 14.5<\/p>\n\n\n\n

So, let’s find y<\/p>\n\n\n\n

T3<\/sub> = y = 11 + (3 – 1)3.5
y = 11 + (2)3.5
y = 11 + 7
Therefore, y = 18<\/p>\n\n\n\n

Series<\/strong><\/h3>\n\n\n\n

A series is obtained when the terms of a sequence are added.
For Example:<\/b>
a) 9 + 12 + 15 + 18 + … + 51
b) 4 + 8 + 12 + .. + 48
c) 1 + 2 + 3 + 4 + …
d) 5 + 10 + 15 + 20 + …<\/p>\n\n\n\n

Examples a) and b) are called finite series<\/b> because they have a definite number of terms and it is always possible to find the sum of a finite series<\/b>.
Examples c) and d) are called infinite series<\/b>. The sum of infinite series is often impossible to find.<\/p>\n\n\n\n

Let’s solve something<\/b>
Find the sum of this A.P.<\/b>
2 + 4 + 6 + … + 98 + 100
Solution<\/strong>
a (First Term) = 2, l (Last Term) = 100
d (Common Difference) = T2<\/sub> – T1<\/sub>
d (Common Difference) = 4 – 2 = 2<\/p>\n\n\n\n

Tn<\/sub> = a + (n – 1)d
100 = 2 + (n – 1)2
100 = 2 + 2n – 2
This means 100 = 2n
n = 100\/2
n = 50<\/p>\n\n\n\n

Therefore, the number of terms in the arithmetic progression is 50<\/b><\/p>\n\n\n\n

Take a closer look at this<\/b><\/p>\n\n\n\n

S (Sum) = 2 + 4 + 6 + … + 98 + 100
Reversing the Series
S (Sum) = 100 + 98 + … + 6 + 4 + 2<\/p>\n\n\n\n

So, by adding the two series vertically we have:<\/b><\/p>\n\n\n\n

2S = 102 + 102 + … + 102 + 102 + 102
2S = 102 x 50
Hence, 2S = 5100
S = 5100<\/sup>\u20442<\/sub>
S = 2550<\/p>\n\n\n\n

Therefore, the sum of the arithmetic series is 2550<\/b><\/p>\n\n\n\n

The following expression represents a general arithmetic series, where the terms are added.<\/p>\n\n\n\n

S = a + (a + d) + (a + 2d) + … + l … (III)
S = l + (l – d) + (l – 2d) + … + a … (IV)<\/p>\n\n\n\n

So, by adding the equations (III) and (IV)<\/h5>\n\n\n\n

2S = (a + l) + (a + l) + … + (a + l)
2S = (a + l)n
S = n(a + l)<\/sup>\u20442<\/sub> … (V)<\/p>\n\n\n\n

The above formula is used when first term and last term is given<\/p>\n\n\n\n

There is another formula<\/p>\n\n\n\n

Since l = a + (n – 1)d
Then substituting l = a + (n – 1)d into equation (V)
Sn<\/sub> = n(a + l)<\/sup>\u20442<\/sub>
Sn<\/sub> = n(a + a + (n – 1)d)<\/sup>\u20442<\/sub>
Therefore, Sn<\/sub> = n(2a + (n – 1)d)<\/sup>\u20442<\/sub> … (VI)<\/p>\n\n\n\n

The above equation or equation (VI) is used when the first term and common difference is given.<\/p>\n\n\n\n

Let’s take another example<\/strong><\/p>\n\n\n\n

a)<\/b> The first term and last term of an arithmetic progression are 2 and 256 respectively. Hence, how many terms are there in the sequence and what is the common difference of the series if its sum is 1548?<\/p>\n\n\n\n

Solution<\/strong><\/h4>\n\n\n\n

a)
First, Let’s find the number of terms, n
a (First Term) = 2
l (Last Term) = 256
Sn<\/sub> (Sum of the Series) = 1548<\/p>\n\n\n\n

Sn<\/sub> = n(a + l)<\/sup>\u20442<\/sub>
1548 = n(2 + 256)<\/sup>\u20442<\/sub>
Then, multiplying throughout by 2
3096 = n(2 + 256)
3096 = 258n
n = 3096<\/sup>\u2044258<\/sub>
n = 12<\/p>\n\n\n\n

Therefore, the number of terms of the series is 12<\/b><\/p>\n\n\n\n

So, let’s find the common difference, d<\/p>\n\n\n\n

Sn<\/sub> = n(2a + (n – 1)d)<\/sup>\u20442<\/sub>
1548 = 12(2(2) + (12 – 1)d)<\/sup>\u20442<\/sub>
So, multiplying throughout by 2
This will give you 3096 = 12(4 + (11)d)
3096 = 48 + 132d
3096 – 48 = 132d
3048 = 132d
d = 3048<\/sup>\u2044132<\/sub>
d = 23.0909<\/p>\n\n\n\n

Therefore, the common difference is 23.0909<\/b>.<\/p>\n\n\n\n

Nickzom Calculator<\/a> is capable of solving all the parameters of an A<\/strong>rithmetic P<\/strong>rogression.<\/p>\n\n\n\n

Download Our Free Android App<\/a><\/p>\n<\/div><\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"Arithmetic Progression A sequence is a set of numbers arranged in a definite pattern. Each number is called…","protected":false},"author":1,"featured_media":11962,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_opengraph-title":"","_yoast_wpseo_opengraph-description":"","_yoast_wpseo_twitter-title":"","_yoast_wpseo_twitter-description":"","_lmt_disableupdate":"","_lmt_disable":"","_sitemap_exclude":false,"_sitemap_priority":"","_sitemap_frequency":"","_yoast_wpseo_focuskw_text_input":"","csco_display_header_overlay":false,"csco_singular_sidebar":"","csco_page_header_type":"","footnotes":"","_members_access_role":[],"_members_access_error":""},"categories":[4],"tags":[10,15,13,14,17,11,12,16],"class_list":["post-60","post","type-post","status-publish","format-standard","has-post-thumbnail","category-mathematics","tag-arithmetic-progression","tag-common-difference","tag-first-term","tag-last-term","tag-number-of-terms","tag-sequence","tag-series","tag-sum-of-a-series","cs-entry"],"yoast_head":"\nHow To Calculate and Solve Arithmetic Progression In Sequences and Series<\/title>\n<meta name=\"description\" content=\"Here is a simple guide on how to easily calculate and solve Arithmetic Progression In Sequences and Series.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How To Calculate and Solve Arithmetic Progression In Sequences and Series\" \/>\n<meta property=\"og:description\" content=\"Here is a simple guide on how to easily calculate and solve Arithmetic Progression In Sequences and Series.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/\" \/>\n<meta property=\"og:site_name\" content=\"Nickzom Academy\" \/>\n<meta property=\"article:published_time\" content=\"2017-09-05T08:21:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-02-05T14:41:56+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2017\/09\/ca46f3c7-20d4-4a3f-b294-005b591f869b.jpeg\" \/>\n\t<meta property=\"og:image:width\" content=\"1024\" \/>\n\t<meta property=\"og:image:height\" content=\"1024\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Edu Guide\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Edu Guide\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"9 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/\"},\"author\":{\"name\":\"Edu Guide\",\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/#\\\/schema\\\/person\\\/5136d132db2835b87874fea743dbffa6\"},\"headline\":\"How To Calculate and Solve Arithmetic Progression In Sequences and Series\",\"datePublished\":\"2017-09-05T08:21:00+00:00\",\"dateModified\":\"2024-02-05T14:41:56+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/\"},\"wordCount\":1249,\"commentCount\":26,\"publisher\":{\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/wp-content\\\/uploads\\\/2017\\\/09\\\/ca46f3c7-20d4-4a3f-b294-005b591f869b.jpeg\",\"keywords\":[\"Arithmetic Progression\",\"Common Difference\",\"First Term\",\"Last Term\",\"Number of Terms\",\"Sequence\",\"Series\",\"Sum of A Series\"],\"articleSection\":[\"Mathematics\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/\",\"url\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/\",\"name\":\"How To Calculate and Solve Arithmetic Progression In Sequences and Series\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/wp-content\\\/uploads\\\/2017\\\/09\\\/ca46f3c7-20d4-4a3f-b294-005b591f869b.jpeg\",\"datePublished\":\"2017-09-05T08:21:00+00:00\",\"dateModified\":\"2024-02-05T14:41:56+00:00\",\"description\":\"Here is a simple guide on how to easily calculate and solve Arithmetic Progression In Sequences and Series.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/#primaryimage\",\"url\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/wp-content\\\/uploads\\\/2017\\\/09\\\/ca46f3c7-20d4-4a3f-b294-005b591f869b.jpeg\",\"contentUrl\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/wp-content\\\/uploads\\\/2017\\\/09\\\/ca46f3c7-20d4-4a3f-b294-005b591f869b.jpeg\",\"width\":1024,\"height\":1024,\"caption\":\"How To Calculate and Solve Arithmetic Progression In Sequences and Series\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/2017\\\/09\\\/05\\\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"How To Calculate and Solve Arithmetic Progression In Sequences and Series\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/#website\",\"url\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/\",\"name\":\"Nickzom Academy\",\"description\":\"Unlock the power of learning: Dive into the world of calculations and excel in school education with Nickzom.\",\"publisher\":{\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/#organization\",\"name\":\"Nickzom Academy\",\"url\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/wp-content\\\/uploads\\\/2023\\\/06\\\/icon-1024.png\",\"contentUrl\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/wp-content\\\/uploads\\\/2023\\\/06\\\/icon-1024.png\",\"width\":1024,\"height\":1024,\"caption\":\"Nickzom Academy\"},\"image\":{\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/#\\\/schema\\\/logo\\\/image\\\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/#\\\/schema\\\/person\\\/5136d132db2835b87874fea743dbffa6\",\"name\":\"Edu Guide\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/wp-content\\\/uploads\\\/2024\\\/08\\\/cropped-nickzom-user-96x96.png\",\"url\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/wp-content\\\/uploads\\\/2024\\\/08\\\/cropped-nickzom-user-96x96.png\",\"contentUrl\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/wp-content\\\/uploads\\\/2024\\\/08\\\/cropped-nickzom-user-96x96.png\",\"caption\":\"Edu Guide\"},\"sameAs\":[\"https:\\\/\\\/www.nickzom.org\\\/\"],\"url\":\"https:\\\/\\\/www.nickzom.org\\\/blog\\\/author\\\/edu-guide\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"How To Calculate and Solve Arithmetic Progression In Sequences and Series","description":"Here is a simple guide on how to easily calculate and solve Arithmetic Progression In Sequences and Series.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/","og_locale":"en_US","og_type":"article","og_title":"How To Calculate and Solve Arithmetic Progression In Sequences and Series","og_description":"Here is a simple guide on how to easily calculate and solve Arithmetic Progression In Sequences and Series.","og_url":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/","og_site_name":"Nickzom Academy","article_published_time":"2017-09-05T08:21:00+00:00","article_modified_time":"2024-02-05T14:41:56+00:00","og_image":[{"width":1024,"height":1024,"url":"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2017\/09\/ca46f3c7-20d4-4a3f-b294-005b591f869b.jpeg","type":"image\/jpeg"}],"author":"Edu Guide","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Edu Guide","Est. reading time":"9 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/#article","isPartOf":{"@id":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/"},"author":{"name":"Edu Guide","@id":"https:\/\/www.nickzom.org\/blog\/#\/schema\/person\/5136d132db2835b87874fea743dbffa6"},"headline":"How To Calculate and Solve Arithmetic Progression In Sequences and Series","datePublished":"2017-09-05T08:21:00+00:00","dateModified":"2024-02-05T14:41:56+00:00","mainEntityOfPage":{"@id":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/"},"wordCount":1249,"commentCount":26,"publisher":{"@id":"https:\/\/www.nickzom.org\/blog\/#organization"},"image":{"@id":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/#primaryimage"},"thumbnailUrl":"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2017\/09\/ca46f3c7-20d4-4a3f-b294-005b591f869b.jpeg","keywords":["Arithmetic Progression","Common Difference","First Term","Last Term","Number of Terms","Sequence","Series","Sum of A Series"],"articleSection":["Mathematics"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/","url":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/","name":"How To Calculate and Solve Arithmetic Progression In Sequences and Series","isPartOf":{"@id":"https:\/\/www.nickzom.org\/blog\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/#primaryimage"},"image":{"@id":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/#primaryimage"},"thumbnailUrl":"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2017\/09\/ca46f3c7-20d4-4a3f-b294-005b591f869b.jpeg","datePublished":"2017-09-05T08:21:00+00:00","dateModified":"2024-02-05T14:41:56+00:00","description":"Here is a simple guide on how to easily calculate and solve Arithmetic Progression In Sequences and Series.","breadcrumb":{"@id":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/#primaryimage","url":"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2017\/09\/ca46f3c7-20d4-4a3f-b294-005b591f869b.jpeg","contentUrl":"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2017\/09\/ca46f3c7-20d4-4a3f-b294-005b591f869b.jpeg","width":1024,"height":1024,"caption":"How To Calculate and Solve Arithmetic Progression In Sequences and Series"},{"@type":"BreadcrumbList","@id":"https:\/\/www.nickzom.org\/blog\/2017\/09\/05\/how-to-calculate-and-solve-arithmetic-progression-in-sequences-and-series\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.nickzom.org\/blog\/"},{"@type":"ListItem","position":2,"name":"How To Calculate and Solve Arithmetic Progression In Sequences and Series"}]},{"@type":"WebSite","@id":"https:\/\/www.nickzom.org\/blog\/#website","url":"https:\/\/www.nickzom.org\/blog\/","name":"Nickzom Academy","description":"Unlock the power of learning: Dive into the world of calculations and excel in school education with Nickzom.","publisher":{"@id":"https:\/\/www.nickzom.org\/blog\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.nickzom.org\/blog\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/www.nickzom.org\/blog\/#organization","name":"Nickzom Academy","url":"https:\/\/www.nickzom.org\/blog\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.nickzom.org\/blog\/#\/schema\/logo\/image\/","url":"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2023\/06\/icon-1024.png","contentUrl":"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2023\/06\/icon-1024.png","width":1024,"height":1024,"caption":"Nickzom Academy"},"image":{"@id":"https:\/\/www.nickzom.org\/blog\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/www.nickzom.org\/blog\/#\/schema\/person\/5136d132db2835b87874fea743dbffa6","name":"Edu Guide","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2024\/08\/cropped-nickzom-user-96x96.png","url":"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2024\/08\/cropped-nickzom-user-96x96.png","contentUrl":"https:\/\/www.nickzom.org\/blog\/wp-content\/uploads\/2024\/08\/cropped-nickzom-user-96x96.png","caption":"Edu Guide"},"sameAs":["https:\/\/www.nickzom.org\/"],"url":"https:\/\/www.nickzom.org\/blog\/author\/edu-guide\/"}]}},"modified_by":"Joshua Abu","_links":{"self":[{"href":"https:\/\/www.nickzom.org\/blog\/wp-json\/wp\/v2\/posts\/60","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.nickzom.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.nickzom.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.nickzom.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.nickzom.org\/blog\/wp-json\/wp\/v2\/comments?post=60"}],"version-history":[{"count":0,"href":"https:\/\/www.nickzom.org\/blog\/wp-json\/wp\/v2\/posts\/60\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.nickzom.org\/blog\/wp-json\/wp\/v2\/media\/11962"}],"wp:attachment":[{"href":"https:\/\/www.nickzom.org\/blog\/wp-json\/wp\/v2\/media?parent=60"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.nickzom.org\/blog\/wp-json\/wp\/v2\/categories?post=60"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.nickzom.org\/blog\/wp-json\/wp\/v2\/tags?post=60"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}