Question

Sum of coefficients of terms of odd powers of x in (1-x+x^2-x^3)^9 is
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Answer 1

Answer:

Let

(1−x+x^2−x^3)^9=a0+a1x+a2x2+⋯+a26x26+a27x27.

If you plug in x=1, then you get

0=a0+a1+a2+⋯+a26+a27. (1)

If you plug in x=−1, then you get

49=a0−a1+a2−⋯+a26−a27. (2)

Now take equation (1) and subtract equation (2) to get

−49=2a1+2a3+⋯+2a25+2a27.

Finally, just divide both sides by 2 to get

−492=a1+a3+⋯+a25+a27.

So, the sum of the coefficients of the odd powers of x is −492=−131072.

Answer 2

Answer:

Let

(1−x+x2−x3)9=a0+a1x+a2x2+⋯+a26x26+a27x27.

now,

If you plug in x=1, then you get

0=a0+a1+a2+....+a25+a26+a27. (1)

If you plug in x=−1, then you get

49=a0−a1+a2−⋯+a26−a27. (2)

Now take equation (1) and subtract equation (2) to get

−49=2a1+2a3+⋯+2a25+2a27.

now divide both sides with 2

−492=a1+a....+a25+a27.

So, the sum of the coefficients of the odd powers of x is −492=−131072.

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