JEE Main 2024 — 29 January, Shift 1 — Question 1

Let the Geometric Progression (G.P.) with 64 terms be:

$$a, ar, ar^2, ar^3, \dots, ar^{63}$$

. Sum of all 64 terms ($S_{64}$) The sum of all terms is given by:

$$S_{64} = \frac{a(r^{64} - 1)}{r - 1}$$

Sum of the odd terms ($S_{odd}$) The odd terms are the 1st, 3rd, 5th, \dots, 63rd terms:

$$a, ar^2, ar^4, \dots, ar^{62}$$

This is a G.P. with: First term = $a$ Common ratio = $r^2$ Number of terms = $\frac{64}{2} = 32$ The sum is:

$$S_{odd} = \frac{a((r^2)^{32} - 1)}{r^2 - 1} = \frac{a(r^{64} - 1)}{r^2 - 1}$$

Setting up the Equation According to the problem, $S_{64} = 7 \times S_{odd}$:

$$\frac{a(r^{64} - 1)}{r - 1} = 7 \times \frac{a(r^{64} - 1)}{r^2 - 1}$$

Solving for $r$ Assuming $a \neq 0$ and $r \neq 1$, we cancel common terms:

$$\frac{1}{r - 1} = \frac{7}{r^2 - 1}$$

Using the identity $r^2 - 1 = (r - 1)(r + 1)$:

$$\frac{1}{r - 1} = \frac{7}{(r - 1)(r + 1)}$$ $$1 = \frac{7}{r + 1}$$ $$r + 1 = 7$$ $$r = 6$$