The sum of the first two terms of a geometric | vedclass

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(B) Let the first term be $a$ and the common ratio be $r$.The terms are $a, ar, ar^2, ar^3, \dots$Given $a + ar = 12$ and $ar^2 + ar^3 = 48$.Factoring

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$-12$

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(B) Let the first term be $a$ and the common ratio be $r$.The terms are $a, ar, ar^2, ar^3, \dots$Given $a + ar = 12$ and $ar^2 + ar^3 = 48$.Factoring the second equation: $r^2(a + ar) = 48$.Substituting $a + ar = 12$ into the equation: $12r^2 = 48$.$r^2 = 4$,so $r = \pm 2$.Since the terms alternate between positive and negative,the common ratio $r$ must be negative. Thus,$r = -2$.Substituting $r = -2$ into $a + ar = 12$:$a + a(-2) = 12$$a - 2a = 12$$-a = 12$$a = -12$.

The sum of the first two terms of a geometric progression is $12$. The sum of the third and fourth terms is $48$. The terms of the geometric progression alternate between positive and negative. What is the first term?

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