$(-1)^{-10} + (-1)^{-9} + (-1)^{-8} + \cdots + (-1)^9 + (-1)^{10}$ (The dots $\cdots$ mean that there are 21 numbers being added, one for each integer from $-10$ to 10.) ?

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$(-1)^{-10} + (-1)^{-9} + (-1)^{-8} + \cdots + (-1)^9 + (-1)^{10}$ (The dots $\cdots$ mean that there are 21 numbers being added, one for each integer from $-10$ to 10.) ?

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Answer 1 · 2018-07-23T22:45:55

The sum of the sequence is $1$

Explanation:

Logically, if we are adding $1$ and $-1$ repeatedly, the sum is $0$ , but since the first and last terms of the sequence are both $1$ , we know that in the sequence there is one more $1$ than $-1$ .

We can prove it with a geometric sum formula for finite sums:
$S_n= a_1((1-r^n)/(1-r))$

$S_21= 1((1-(-1)^(21))/(1-(-1)))$

$S_21= 1(2/2)$

$S_21= 1$

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$(-1)^{-10} + (-1)^{-9} + (-1)^{-8} + \cdots + (-1)^9 + (-1)^{10}$ (The dots $\cdots$ mean that there are 21 numbers being added, one for each integer from $-10$ to 10.) ?

1 Answer

Explanation:

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(-1)^{-10} + (-1)^{-9} + (-1)^{-8} + \cdots + (-1)^9 + (-1)^{10}(-1)^{-10} + (-1)^{-9} + (-1)^{-8} + \cdots + (-1)^9 + (-1)^{10} (The dots \cdots\cdots mean that there are 21 numbers being added, one for each integer from -10-10 to 10.) ?

1 Answer

Explanation:

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$(-1)^{-10} + (-1)^{-9} + (-1)^{-8} + \cdots + (-1)^9 + (-1)^{10}$ (The dots $\cdots$ mean that there are 21 numbers being added, one for each integer from $-10$ to 10.) ?