Single Choice

An infinite G.P. has first term as a and sum as $$5$$, then?

A$$a < -10$$
B$$-10 < a < 10$$
C$$0 < a < 10$$ and $$a\neq 15$$
Correct Answer
D$$a > 10$$

Solution

$$S_{\infty} = \dfrac{a}{1-r} = 5$$ (given)

$$\Rightarrow r = \dfrac{5-a}{5}$$
But
$$0 < |r| < 1$$

$$\Rightarrow 0 < \Big| \dfrac{5-a}{5} \Big| < 1$$

$$\Rightarrow -1 < \dfrac{5-1}{5} < 1 \ and \ a \neq 5$$
$$\Rightarrow -5 < 5 - a < 5 \ and \ a \neq 5$$
$$\Rightarrow -10 < -a < 0 \ and \ a \neq 5$$
$$\Rightarrow 10 > a > 0 \ and \ a \neq 5$$
$$\Rightarrow 0 < a < 10 \ and \ a \neq 5$$

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