{"id":5406,"date":"2019-07-29T10:10:05","date_gmt":"2019-07-29T04:40:05","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=5406"},"modified":"2022-11-04T12:06:29","modified_gmt":"2022-11-04T06:36:29","slug":"common-difference-of-arithmetic-progression-or-sequence","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/common-difference-of-arithmetic-progression-or-sequence\/","title":{"rendered":"common difference of arithmetic progression or sequence"},"content":{"rendered":"

Common difference of arithmetic progression or sequence is defined as the difference between the two consecutive terms in the arithmetic progression or sequence or series.\u00a0 It is constant across the whole series<\/p>\n

if $a_1$,$a_2$ ,$a_3$.$a_4$ …..\u00a0 are in arithmetic progression or sequence<\/p>\n

then<\/p>\n

$d=a_2 -a_1 = a_3 -a_2 =a_4 -a_3$….<\/p>\n

If the common difference is not same across the series ,then series is not in Arithmetic progression. This is important condition for the series to be in A.P.<\/p>\n

So common difference of the Arithmetic series can be found\u00a0 given the two consecutive terms in the A.P<\/p>\n

Examples<\/strong><\/p>\n

Find the common difference of the below\u00a0 Arithmetic progression<\/p>\n

a. $2 + \\sqrt {3}$ , $2+\u00a0 \\sqrt {12}$ , $2+\u00a0 \\sqrt {27}$, ,$2 + \\sqrt {48}$ ….<\/p>\n

b.\u00a0 -10 , -7 , -4 ,-1 ,2…….<\/p>\n

c. 2, 3.5 , 5, 6.5 , 8……..<\/p>\n

d.\u00a0 1 , 3\/2 , 2 , 5\/2 ……<\/p>\n

Solutions<\/strong><\/p>\n

a.<\/p>\n

Here $d= 2+\u00a0 \\sqrt {12} – 2 + \\sqrt {3} = \\sqrt {12} – \\sqrt {3} = \\sqrt {4 \\times 3} – \\sqrt {3} = 2 \\sqrt {3} – \\sqrt {3}= \\sqrt {3}$<\/p>\n

$d= 2+\u00a0 \\sqrt {27} – 2+\u00a0 \\sqrt {12} = 3 \\sqrt {3} – 2 \\sqrt {3}= \\sqrt {3}$<\/p>\n

So common difference is $\\sqrt {3}$<\/p>\n

b.\u00a0 d= -7 – (-10) = 3<\/p>\n

d= -4 – (-7) = 3<\/p>\n

…..<\/p>\n

So common difference is 3<\/p>\n

c.\u00a0 d= 3.5 -2 = 1.5<\/p>\n

d= 5 -3.5 = 1.5<\/p>\n

……<\/p>\n

So common difference is 1.5<\/p>\n

d.\u00a0 $d = \\frac {3}{2} -1\u00a0 =\\frac {1}{2}$<\/p>\n

$d= 2 – \\frac {3}{2} = \\frac {1}{2}$<\/p>\n

……<\/p>\n

So common difference is 1\/2<\/p>\n

How to find the common difference if the nth term of the Arithmetic Progression is given<\/strong><\/p>\n

Steps would be<\/p>\n

a. First find the first term by putting n=1<\/p>\n

b. Find the second term by putting n=2<\/p>\n

Now common\u00a0 difference = Second term – First term<\/p>\n

Examples<\/strong><\/p>\n

a. Nth term is given as<\/p>\n

$T_n = 2n -1$<\/p>\n

b. Nth term is given as<\/p>\n

$T_n = 4n -3$<\/p>\n

Solutions<\/strong><\/p>\n

a.<\/p>\n

$T_n = 2n -1$<\/p>\n

$T_1 = 2 \\times 1 – 1 = 1$<\/p>\n

$T_2 = 2 \\times 2 -1 = 3$<\/p>\n

Common difference = 3 -1 =2<\/p>\n

b.<\/p>\n

$T_n = 4n -3$<\/p>\n

$T_1 = 4 \\times 1 – 3 = 1$<\/p>\n

$T_2 = 4 \\times 2 – 3 = 5$<\/p>\n

Common difference = 5 -1 = 4<\/p>\n

How to find the common difference if sum of the nth term of the Arithmetic Progression is given<\/strong><\/p>\n

Steps would be<\/p>\n

a. First\u00a0 sum up-to to first term by putting n=1\u00a0 ($S_1$)<\/p>\n

b. Find sum up-to to second term by putting n=2 ( $S_2$)<\/p>\n

Now second\u00a0 Term= $S_2 – S_1$<\/p>\n

Common difference = Second term – First term = $S_2 – S_1 – S_1 = S_2 – 2S_1$<\/p>\n

Examples<\/strong><\/p>\n

$S_n\u00a0 = 3 n^2 + 5n $<\/p>\n

Solution<\/strong><\/p>\n

$S_1 = 3 (1)^2 + 5 \\times 1 = 8$<\/p>\n

$S_2 = 3 (2)^2 + 5 \\times 2 =22$<\/p>\n

Now second term = $S_2 – S_1 = 22 -8=14$<\/p>\n

Common difference = Second term – First term = $S_2 – S_1 – S_1$ = $22 – 2 \\times 8=6$<\/p>\n

How to find the common difference if two terms are given in the Arithmetic Series<\/strong><\/p>\n

Suppose we have 10th and 16th terms given as<\/p>\n

$T_{10}= 20$<\/p>\n

$T_{16}=32$<\/p>\n

Let a\u00a0 be the first term and d be the common difference<\/p>\n

then using nth term formula<\/p>\n

$a+ (10-1)d= 20$ or $a+9d =20$<\/p>\n

$a+ (16-1)d =32$ or $a + 15 d=32$<\/p>\n

Solving these\u00a0 we get a=2 ,d=2<\/p>\n

I hope you find this article on common difference of arithmetic progression or sequence useful<\/p>\n

Related Articles<\/strong><\/p>\n

What is Arithmetic Progression<\/a><\/p>\n

Arithmetic Progressions Worksheet<\/a><\/p>\n

Important Questions for Arithmetic Progression<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Common difference of arithmetic progression or sequence is defined as the difference between the two consecutive terms in the arithmetic progression or sequence or series.\u00a0 It is constant across the whole series if $a_1$,$a_2$ ,$a_3$.$a_4$ …..\u00a0 are in arithmetic progression or sequence then $d=a_2 -a_1 = a_3 -a_2 =a_4 -a_3$…. If the common difference is […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-5406","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nHow to find common difference in Arithmetic Progressions or Series<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/common-difference-of-arithmetic-progression-or-sequence\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How to find common difference in Arithmetic Progressions or Series\" \/>\n<meta property=\"og:description\" content=\"Common difference of arithmetic progression or sequence is defined as the difference between the two consecutive terms in the arithmetic progression or sequence or series.\u00a0 It is constant across the whole series if $a_1$,$a_2$ ,$a_3$.$a_4$ …..\u00a0 are in arithmetic progression or sequence then $d=a_2 -a_1 = a_3 -a_2 =a_4 -a_3$…. 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difference of arithmetic progression or sequence is defined as the difference between the two consecutive terms in the arithmetic progression or sequence or series.\u00a0 It is constant across the whole series if $a_1$,$a_2$ ,$a_3$.$a_4$ …..\u00a0 are in arithmetic progression or sequence then $d=a_2 -a_1 = a_3 -a_2 =a_4 -a_3$…. 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