An infinite geometric series has a sum of 20, where all the terms are positive. The sum of the first and second terms are 12.8. What is the first term?

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An infinite geometric series has a sum of 20, where all the terms are positive. The sum of the first and second terms are 12.8. What is the first term?

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Answer 1 · 2018-03-30T16:11:03

$a = 8$ $a = 8$

Explanation:

Recall the sum of the terms in an infinite geometric series is $20$ $20$ .

$S_{n} = \frac{a}{1 - r}$ $S_n = a/(1 - r)$

We know the sum of the series, so:

$20 = \frac{a}{1 - r}$ $20 = a/(1 - r)$

$a = 20 (1 - r)$ $a = 20(1 - r)$

The first term is $a$ $a$ . The second term is $a r$ $ar$ . Therefore, $s_{2} = a + a r = a (1 + r)$ $s_2 = a + a r = a(1 + r)$ . We know the sum of the first two terms is $12.8 = \frac{64}{5}$ $12.8 = 64/5$ . Therefore our second equation becomes $a (1 + r) = \frac{64}{5}$ $a(1 + r) = 64/5$ .

Substituting the first equation into the second we get:

$(20 (1 - r)) (1 + r) = \frac{64}{5}$ $(20(1 - r))(1 +r) = 64/5$

$(20 - 20 r) (1 + r) = \frac{64}{5}$ $(20 - 20r)(1 + r ) = 64/5$

$20 - 20 r + 20 r - 20 r^{2} = \frac{64}{5}$ $20 - 20r + 20r - 20r^2 = 64/5$

$100 - 100 r^{2} = 64$ $100 - 100r^2 = 64$

$36 = 100 r^{2}$ $36 = 100r^2$

$0.36 = r^{2}$ $0.36 = r^2$

$r = \pm 0.6$ $r = +- 0.6$

Since all the terms are positive, $r = + 0.6$ $r = +0.6$ (if the common ratio was negative, every two terms would be negative).

Since $a = 20 (1 - r)$ $a = 20(1 - r)$ , $a = 20 (0.4) = 8$ $a = 20(0.4) = 8$

Hopefully this helps!

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An infinite geometric series has a sum of 20, where all the terms are positive. The sum of the first and second terms are 12.8. What is the first term?

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