Algebra

Progression & Series

Medium

The sum of the first 5 terms of a geometric progression is the same as the sum of the first 7 terms of the same progression. If the sum of the first 9 terms is 24, then the 4th term of the progression is

-48

-24

✅ Correct Option: 2

We have a Geometric Progression (GP) where the sum of the first 5 terms equals the sum of the first 7 terms, the sum of the first 9 terms is 24, and we need to find the 4th term.

The sum of first $n$ terms of a GP is:

$S_n = a \cdot \dfrac{1-r^n}{1-r}$

where $a$ is the first term and $r$ is the common ratio.

Since $S_5 = S_7$ , we can write:

$S_7 - S_5 = 0$

This means the sum of the 6th and 7th terms is zero:

$a_6 + a_7 = 0$

$ar^5 + ar^6 = 0$

$ar^5(1 + r) = 0$

Since $a \neq 0$ and $r \neq 0$ , we get:

$r = -1$

When $r = -1$ , the terms of the GP alternate between $a$ and $-a$ :

$a, \ -a, \ a, \ -a, \ a, \ -a, \ a, \ -a, \ a$

Adding up the first 9 terms:

$S_9 = (a - a) + (a - a) + (a - a) + (a - a) + a = a$

We are given $S_9 = 24$ , so:

$a = 24$

The 4th term of a GP is $a_4 = ar^{3}$ :

$a_4 = 24 \times (-1)^3$

$a_4 = -24$

Verification with $a = 24$ and $r = -1$ :

GP: $24, \ -24, \ 24, \ -24, \ 24, \ -24, \ 24, \ -24, \ 24$

$S_5 = 24$ , $S_7 = 24$ , so $S_5 = S_7$ ✓

$S_9 = 24$ ✓

The 4th term of the progression is $-24$ .

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