Find the coefficient of x50 in the polynomials after parenthesis have been removed and like terms have been collected in the expansion(1+x) 1000+x(1+x) 999+x2(1+x) 998+....+x1000

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Find the coefficient of x50 in the polynomials after parenthesis have been removed and like terms have been collected in the expansion(1+x) 1000+x(1+x) 999+x2(1+x) 998+....+x1000

Filo · Accepted Answer

Given expansion is (1+x) 1000+2x(1+x) 999+3x2(1+x) 998+3x3(1+x) 997+....+1001x1000Let S denotes the sum of the polynomial. S=(1+x) 1000+2x(1+x) 999+3x2(1+x) 998+3x3(1+x) 997+....+1001x1000             .....(1)Multiplying by 1+xx​ in eqn (1), we get (1+x)x​S=       x(1+x) 999+2x2(1+x) 998+3x3(1+x) 997+...+1000x1000+(1+x)1001​x1001      ....(2)Subtracting eqn(2) from eqn(1), we get(1−1+xx​)S=(1+x)1000+x(1+x) 999+x2(1+x) 998+x3(1+x) 997+...+x1000−(1+x)1001​x1001⇒(1+x1​)S=(1+x)1000+x(1+x) 999+x2(1+x) 998+x3(1+x) 997+....+x1000−(1+x)1001​x1001⇒ S=[(1+x)1001+x(1+x) 1000+x2(1+x) 999+x3(1+x) 998+....+x1000(1+x)]−1001x1001The sequence in square bracket forms a G.P. Here, first term a=(1+x)1001 and common ratio r=1+xx​.So, using the formula for sum of n terms of G.P. Sn​=1−ra(1−rn)​⇒S=1−1+xx​ (1+x)1001[1−(1+xx​ )1001]​−1001x1001⇒S=1+x1​ (1+x)1001[1−(1+xx​ )1001]​−1001x1001⇒S=(1+x)1002[1−(1+xx​ )1001]−1001x1001⇒S=(1+x)1002−(1+x)x1001−1001x1001⇒S=(1+x)1002−1002x1001−x1002Since, the last two terms in the above sum cannot have x50 and only the first term can have x50Now, the coefficient of xr in the expansion of (1+x)n is nCr​So, the coefficient of x50 in (1+x)1002 is 1002C50​

Question Text	Find the coefficient of $x^{50}$ in the polynomials after parenthesis have been removed and like terms have been collected in the expansion $(1 + x)^{1000} + x (1 + x)^{999} + x^{2} (1 + x)^{998} + .... + x^{1000}$
Answer Type	Text solution:1
Upvotes	150

Find the coefficient of $x^{50}$ in the polynomials after parenthesis have been removed and like terms have been collected in the expansion
$(1 + x)^{1000} + x (1 + x)^{999} + x^{2} (1 + x)^{998} + .... + x^{1000}$

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