Question

The sum of how many terms of an A.P.: 7, 11, 15, 19, 23, ….. will be 900?

Answer 1

Hint: Assume the number of terms to be n. Solve for n using the formula $sum=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$. Here a = 7, d = 4 and sum = 900.

Complete step-by-step answer:
Given, A. P. is:
7, 11, 15, 19, 23, …..
In the given A.P. the first term is 7 and the common difference is (15-11) = 4.
We know that the sum of an A.P.
$S=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$
where n, a and d are the number of terms, first term and common difference respectively.
Given, sum of terms = 900.
We use the above equation for a = 7, d = 4 and S = 900.
Hence, we solve for n i.e. number of terms,
$\begin{aligned} & \Rightarrow 900=\dfrac{n}{2}\left[ 2\times 7+\left( n-1 \right)4 \right] \\\ & \Rightarrow 900=n\left[ 7+\left( n-1 \right)2 \right] \\\ & \Rightarrow 900=n\left( 2n+5 \right) \\\ & \Rightarrow 2{{n}^{2}}+5n-900=0 \\\ \end{aligned}$
We factorize the above equation.
$\begin{aligned} & 2{{n}^{2}}-40n+45n-900=0 \\\ & 2n\left( n-20 \right)+45\left( n-20 \right)=0 \\\ & \Rightarrow \left( 2n+45 \right)\left( n-20 \right)=0 \\\ & n=\dfrac{-45}{2}\ or\ 20. \\\ \end{aligned}$
Since, n is a whole number, it can’t be $\dfrac{-45}{2}$. Therefore, n is 20.
Answer is 20.

Note: Here sequence is given as A.P. so we can directly find common difference as (second term - first term) and if the sequence is not given as A.P. then we have to prove that first then we can find the common difference of A.P.