CAT 2025

Slot 3

QA

Question & Solution

Let the first term of the arithmetic progression be $$a$$, and the common difference be $$d$$.

We have:

$$(a+3d) + (a+6d) + (a+9d) = 99$$

$$\Rightarrow 3a + 18d = 99$$  or  $$a + 6d = 33$$ .....(1)

We are also told that the sum of the first fourteen terms is $$497$$,

$$\dfrac{14}{2}(2a + (14-1)d) = 497$$

$$\Rightarrow (2a+13d) = \dfrac{497}{7}$$

$$\Rightarrow 2a+13d = 71$$ .....(2)

Solving equations (1) and (2), we get;

$$a= 3$$ and $$d=5$$

The first five terms would therefore be: $$3, 8, 13, 18, 23$$, and their sum would be $$3+8+13+18+23 = 65$$.