The ____ is the G.P for which the sum of the first two terms is −4 and the fifth term is 4 times the third term.

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By the given data we have the fifth term is 4 times the third term. The general form of the G.P is given by a, ar, ar2, ar3,... where the nth term is defined as an = arn−1 . Therefore, the fifth term is expressed as a5 = ar5−1 ⇒a5 = ar4 and the third term is expressed as a3 = ar3−1⇒ a3 = ar2 . By the given data we have the fifth term is 4 times the third term so we have,
a5 = 4a3
⇒ar4 = 4ar2
On simplification we have,

\Rightarrow \frac{a r^{4}}{a r^{2}} = 4

By canceling the same quantity, we have,
⇒ r2 = 4
Applying the square root, we have
r = ±2
We have found the common ratio and we have to find the first element of the progression.
By the given data we have the sum of the first two terms is -4.
We have formula for the sum of the G.P we have two kind of depending on the common ratio and that is defined as-
If the common ratio greater than 1 we have

S n = \frac{a (r^{n} - 1)}{r - 1}

and if the common ratio is less than 1 we have

S n = \frac{a ({1 - r}^{n})}{1 - r}

Suppose r > 1 that is r = 2 we have

S n = \frac{a (r^{n} - 1)}{r - 1}

\Rightarrow S 2 = \frac{a (2^{2} - 1)}{2 - 1}

Since S2 = −4 we have
⇒ −4 = a(4−1)1
⇒ −4 = 3a

\Rightarrow a = - \frac{4}{3}

So, we have the first term

a = - \frac{4}{3}

and r = 2 then the geometric progression is

- \frac{4}{3}, - \frac{8}{3}, - \frac{16}{3}, . . . .

So, the correct answer is “

- \frac{4}{3}, - \frac{8}{3}, - \frac{16}{3}, . . . .

”.
Suppose r < 1 that is r = −2 we have

S n = \frac{a ({1 - r}^{n})}{1 - r}

S 2 = \frac{a ({1 - (- 2)}^{2})}{1 - (- 2)}

Since S2=−4 we have

\Rightarrow - 4 = \frac{a (1 - 4)}{3}

\Rightarrow - 4 = - \frac{3 a}{3}

⇒ a = 4
So we have the first term a = 4 and r = −2 then the geometric progression is 4,−8,16,−32,....
So, the correct answer is “4,−8,16,−32,.... ”.

The ____ is the G.P for which the sum of the first two terms is −4 and the fifth term is 4 times the third term.

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