Find the coefficient of x-10 in the expansion: (2-1/x2)8
ANS: -448
I've tried using the General Term formula and got stuck at -x2r = x-10. Also, I tried expanding the equation but it doesn't look like its leading me somewhere.
Find the coefficient of a negative indexed $x$ in a series expansion
binomial theorembinomial-coefficients
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Best Answer
You've got the right idea. We know $$(a+b)^8=\sum_{k=0}^{8}{8\choose k}a^{8-k}b^k$$ so we just need to see what happens when $a=2, b=-x^{-2}.$ If order to get the $x^{-10}$ term, we need $k=5$ since $(x^{-2})^5=x^{10}.$ Then the coefficient will be $${8\choose5}2^{8-5}(-1)^5=-8\cdot56=-448$$