The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is

Sequences and Series PYQ — JEE Main 2008 · Shift 0 · Q48 with Solution

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Correct answer: B

Let the first term and common ratio be $a$ and $r$ .

Then the GP is: $a,\ ar,\ ar^2,\ ar^3,\dots$
Use the given sums.

First two terms add to $12$ : $a+ar=a(1+r)=12 \quad ...(1)$

Third and fourth terms add to $48$ : $ar^2+ar^3=ar^2(1+r)=48 \quad ...(2)$
Relate equations (1) and (2).

From (1), $a(1+r)=12$ .

From (2), $ar^2(1+r)=48$

Divide (2) by (1): $r^2=\frac{48}{12}=4$ $r=\pm 2$
Use the condition: terms are alternately positive and negative.

For signs to alternate in a GP, the common ratio must be negative.

Hence, $r=-2$
Find the first term.

Substitute into (1): $a(1-2)=12$ $a(-1)=12$ $a=-12$
Check:

GP: $-12, 24, -48, 96, \dots$

First two terms: $-12+24=12$

Third and fourth terms: $-48+96=48$

Signs are alternating, so this is correct.

Therefore, the first term is $\boxed{-12}$ , which corresponds to Option B.