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Find the coefficients of x^(50) in the e...

Find the coefficients of `x^(50)` in the expression `(1+x)^(1000)+2x(1+x)^(999)+3x^2(1+x)^(998)++1001 x^(1000)` .

Text Solution

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The given series is `A.G.P..` Lets first find its sum.
Writing t for `1+x`, let
`S = t^(1000) + 2xt^(999) + 3x^(2)t^(998) + "….." + 1001x^(1000)`
`:. (x//t) . S = xt^(999) + 2x^(2)t^(998) + "…." + 1001x^(1001)//t`
Substracting , we get
`S(1-x//t) = t^(1000) + xt^(999) + x^(2)t^(998) + "....." + x^(1000) - 1001x^(1001)//t`
`= (t^(1000) [1-(x//t)^(1001)])/(1-x//t) - (1001x^(1001))/(t)`
or `S = (1+x)^(1002) - x^(1001) (1+x) - 1001 x^(1001)`
(Putting `t = 1+x` and simplifying)
`:.` Coefficient of `x^(50)` in the expansion `=` coefficient of `x^(500` in the expansin of `(1+x)^(1002) = .^(1002)C_(50)`.
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