How to evaluate the coefficient in this expression?

Question

If the coefficient of $x^r$ in the product of $(1-x+x^2-x^3+......+x^{100})(1+x+x^2+x^3+.....+x^{100})$ is denoted by $T_r$. What is the value of $(T_{99}+T_{101}+T_{103})$

I don't know how to approach the question. Do I have to evaluate all of those T's separately? Or do they simplify into something beautiful?

score 4 · Accepted Answer · answered Mar 29 '18 at 16:11

4

This polynomial is $$\frac{1+x^{101}}{1+x}\frac{1-x^{101}}{1-x} =\frac{1-x^{202}}{1-x^2}=1+x^2+x^4+\cdots+x^{200}.$$ What are $T_{99}$ etc.?

answered Mar 29 '18 at 16:11

Angina Seng

158,341

+1 Thanks for the quick answer. Completely forgot about geometric series. – SmarthBansal Mar 29 '18 at 16:14
This was a mcq in an exam, but none of the options had zero. Do you think it's a mistake? – SmarthBansal Mar 29 '18 at 16:20
2

Another was to see all these coefficients are zero is to observe that the expression is an even function of $x$. @SmarthBansal – Angina Seng Mar 29 '18 at 16:25

How to evaluate the coefficient in this expression?

1 Answers1