If the first term of an infinite geometric se | vedclass

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(C) Let the first term be $a$ and the common ratio be $r$,where $|r| < 1$.The infinite geometric series is $a, ar, ar^2, ar^3, \dots$.The sum of all t

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$1/3$

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(C) Let the first term be $a$ and the common ratio be $r$,where $|r| The infinite geometric series is $a, ar, ar^2, ar^3, \dots$.The sum of all terms subsequent to the first term is $S = ar + ar^2 + ar^3 + \dots = \frac{ar}{1-r}$.According to the problem,the first term is twice the sum of the subsequent terms:$a = 2 	imes \left( \frac{ar}{1-r} ight)$.Dividing both sides by $a$ (assuming $a 
eq 0$):$1 = \frac{2r}{1-r}$.$1 - r = 2r$.$1 = 3r$.$r = 1/3$.

If the first term of an infinite geometric series is twice the sum of all its subsequent terms,what is the common ratio?

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