Can you tell me why the sum of geometric series of -3+15-75+375...,n=8 a positive number, not negative?

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Can you tell me why the sum of geometric series of -3+15-75+375...,n=8 a positive number, not negative?

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Answer 1 · 2017-07-08T07:59:48

See below:

Explanation:

Let's look at the series this way:

We have 8 numbers. They start with $- 3$ $-3$ and increase by multiplying by $- 5$ $-5$ . I'm going to group these 8 numbers in groups of 2:

$- 3, 15$ $-3, 15$
$- 75, 375$ $-75, 375$
$- 1875, 9375$ $-1875, 9375$
$- 46875, 234375$ $-46875 , 234375$

If I add up each group of 2 numbers, I'll get a positive number:

$12$ $12$
$300$ $300$
$7500$ $7500$
$187500$ $187500$

And clearly if I add these 4 numbers up, I'll get a positive number.

Answer 2 · 2017-07-08T08:28:48

The given GP has 1st term $a = - 3$ $a=-3$ and common ratio $r = - 5$ $r=-5$

We know the sum of the series up to n th term is

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

Here $S_{n} = \frac{- 3 (1 - {(- 5)}^{n})}{1 - (- 5)}$ $S_n=(-3(1-(-5)^n))/(1-(-5))$

$\Rightarrow S_{n} = - \frac{1}{2} (1 - {(- 5)}^{n})$ $=>S_n=-1/2(1-(-5)^n)$

$\Rightarrow S_{n} = \frac{1}{2} ({(- 5)}^{n} - 1)$ $=>S_n=1/2((-5)^n-1)$

${(- 5)}^{n}$ $(-5)^n$ is positive for all even integral value of n . So when $n = 8$ $n=8$ it will be positive and hence the sum up to 8 (even) term will be positive.

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Can you tell me why the sum of geometric series of -3+15-75+375...,n=8 a positive number, not negative?

2 Answers

Explanation:

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