Question 34

# 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)

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

$$\Rightarrow$$ $$103 = 4k+7$$

$$\Rightarrow$$ $$k = 24$$