Question

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

Answer 1

Hint – In order to solve this problem we need to find the first term common difference from the series given. Then apply the formula of sum of P to get the number of terms in this problem.

Complete step by step answer:
The given AP is 9, 17, 25,...
Then the common difference,
$
  d = {a_2} - {a_1} = 17 - 9 = 8 \\
  d = {a_3} - {a_2} = 25 - 17 = 8 \\
$
So, d = 8 common difference and the first term is 9.
We know that the sum of n terms can be written as ${S_n} = \dfrac{n}{2}(2a + (n - 1)d)$.
We have given the sum as 636.
So, we can do,
$
   \Rightarrow 636 = \dfrac{n}{2}(2(9) + (n - 1)8) \\
   \Rightarrow 1272 = 18n + 8{n^2} - 8n \\
   \Rightarrow 4{n^2} + 5n - 636 = 0 \\
$
On using the formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ and solving we get,
$
   \Rightarrow n = \dfrac{{ - 5 \pm \sqrt {{5^2} - 4(4)( - 636)} }}{{2(4)}} \\
   \Rightarrow n = 12,\, - 13.25 \\
$
Negative values of a number of terms will be neglected. Then the number of terms needed is 12 to get the sum of 636.
Note – In such a type of problem you have to find the first term, common difference from the given AP. Then solve as per the conditions provided in the problem. Here as it is said that we need to find the number of terms needed to get the sum of 636. So, we have all the unknowns except the number of terms so we used the formula ${S_n} = \dfrac{n}{2}(2a + (n - 1)d)$ and got the answer to this question.