Question

The sum of first 7terms of ap is 63 and sum of next seven terma is 161 find 28th term

Answer 1

Arithmetic Progression (AP)

• It's general formula = a, a+ d, a + 2d,....

• In an AP :

a is the first term and d is common difference.

Common difference is is find by using formula : $a_{2}$ - $a_{1}$, $a_{3}$ - $a_{2}$,....

The nth term of an AP is given as : $a_{n}$ = a + (n - 1)d

• The sum of 1st n terms of an AP :

$S_{n}$ = $\dfrac{n}{2}$ [2a + (n - 1)d]

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☞ The sum of first 7 terms of an AP is 63.

Here

n = 7

$S_{n}$ = 63

$S_{n}$ = $\dfrac{n}{2}$ [2a + (n - 1)d]

=> $S_{7}$ = $\dfrac{7}{2}$ [2a + (7 - 1)d]

=> 63 = $\dfrac{7}{2}$ [2a + 6d]

=> 126 = 7(2a + 6d)

=> $\dfrac{126}{7}$ = 2a + 6d

=> 18 = 2a + 6d

=> 2a + 6d = 18 ________(eq 1)

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☞ Sum of next 7 term is 161.

Sum of 1st 7 term + Sum of next 7 term = Sum of 14 terms.

Here

n = 14

$S_{14}$ = 63 + 161 = 224

$S_{14}$ = $\dfrac{14}{2}$ [2a + (14 - 1)d]

=> 224 = 7(2a + 13d)

=> $\dfrac{224}{7}$ = 2a + 13d

=> 32 = 2a + 13d

=> 2a + 13d = 32 ________(eq 2)

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• Solve (eq 1) and (eq 2) by using elimination method we get;

d = 2

• Put value of d in (eq 1)

=> 2a + 6(2) = 18

=> 2a + 12 = 18

=> 2a = 6

a = 3

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☞ We have to find the 28th term of an AP.

$a_{n}$ = a + (n - 1)d

Here

n = 28

$a_{28}$ = a + (28 - 1)d

=> $a_{28}$ = 3 + 27(2) [value of d and a from above]

=> $a_{28}$ = 3 + 54

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=> $a_{28}$ = 57

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