Applyo - College Application Platform

CAT 2018 Slot 2 QA Question & Solution

AlgebraHard

Question

Let $t_{1},t_{2}$,... be real numbers such that $t_{1}+t_{2}+…+t_{n} = 2n^{2}+9n+13$, for every positive integer $n \geq 2$. If $t_{k}=103$, then k equals

Solution

It is given that $t_{1}+t_{2}+…+t_{n} = 2n^{2}+9n+13$, for every positive integer $n \geq 2$. 

We can say that $t_{1}+t_{2}+…+t_{k} = 2k^{2}+9k+13$   ... (1) 

Replacing k by (k-1) we can say that 

 $t_{1}+t_{2}+…+t_{k-1} = 2(k-1)^{2}+9(k-1)+13$   ... (2)

On subtracting equation (2) from equation (1)

$\Rightarrow$ $t_{k} = 2k^{2}+9k+13 - 2(k-1)^{2}+9(k-1)+13$

$\Rightarrow$ $103 = 4k+7$

$\Rightarrow$ $k = 24$