To Prove : coefficient of x10 in (1 - x2)10: coefficient of x0 in \(\Big(x-\frac{2}{x}\Big)^{10}\) = 1:32
For (1 - x2)10 ,
Here, a = 1, b = -x2 and n = 15
We have formula,
To get coefficient of x10 we must have,
(x)2r = x10
• 2r = 10
• r = 5
Therefore, coefficient of x10 = -(105)
For , \(\Big(x-\frac{2}{x}\Big)^{10}\)
Here, a = x, b = \(\frac{-2}{x}\) and n = 10
We have a formula,
Now, to get coefficient of term independent of x that is coefficient of x0 we must have,
(x)10-2r = x0
• 10 - 2r = 0
• 2r = 10
• r = 5
Therefore, coefficient of x0 = -(510)(2)5
Therefore,
= \(\frac{1}{(2)^5}\)
= \(\frac{1}{32}\)
Hence,
Coefficient of x10 in (1-x2)10: coefficient of x0 in \(\Big(x-\frac{2}{x}\Big)^{10}\) = 1:32