Question

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$\huge{\underline{\mathrm{\green{Question-}}}}$

How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?

Answer 1

AnswEr:

12 terms must be taken so that the sum of given AP is 636.

ExplanaTion:

Given :

• a = 9
• d = 17 - 9 → 8
• $S_n$ = 636

To find :

• Number of terms so that sum is 636.

SoluTion:

We know that,

$\large{\boxed{\sf{\red{Sum\:of\:n\:terms\:=\:\dfrac{n}{2} [2a + (n - 1)d]}}}}$

Putting the values, we get,

: $\implies$ 636 = $\sf{\dfrac{n}{2} [2 \times 9 + (n - 1)(8)]}$

: $\implies$ 636 = $\sf{\dfrac{n}{2} [8n + 10]}$

Solving it, we get,

: $\implies$ $\purple{\sf{4n^2 + 5n - 636 = 0}}$

Splitting middle term, we get,

: $\implies$ $\sf{4n^2 - 48n + 53n - 636 = 0}$

: $\implies$ $\sf{4n(n - 12) + 53(n - 12) = 0}$

: $\implies$ $\sf{(4n + 53)(n - 12) = 0}$

: $\implies$ $\sf{n\:=\:\dfrac{-53}{4}\:and\:12}$

Rejecting the negative value, we get,

: $\implies$ $\blue{\sf{n\:=\:12}}$

Hence, 12 terms must be taken so that the sum of given AP is 636.

Answer 2

Answer:

• AP :- 9, 17, 25...
• n = ?
• Sum = 636

We know that, S(n) = n/2 * (2a + (n - 1)d)

From the AP,

a = 9

d = 17 - 9 = 8

Hence, 636 = n/2 * (2(9) + (n - 1)8)

636 = n/2 * (18 + 8n - 8)

636 = n/2 * (10 + 8n)

636 = 5n + 4n²

4n² + 5n = 636

By splitting the middle term,

4n² - 48n + 53n - 636 = 0

4n (n - 12) + 53 (n - 12) = 0

(4n + 53) (n - 12) = 0

n = -53/4, 12

Now, we will obviously have to reject the negative and the fractional value.

Hence, 12 terms must be taken in the arithmetic progression to get the sum as 636.