Grade 12Algebra
The coefficient of x^n in (1+x+2x^2+3x^3.......nx^n) is?
The coefficient of x^n in (1+x+2x^2+3x^3.......nx^n) is?
1 Answer
Arun
Substitute x with 1/x;
Now the coefficient of x is the coefficient of 1/x.
Hence, when I multiply the two series, we get
( (1 + x +....+ x^n)*(x^n + x^(n - 1) +.....+ n)/x^n )^n
Hence, the coefficient of x^n should be the square root of the constant term.
Since the denominator(inside the outermost power) contains x^n, we need the coefficient of x^n in the numerator which is
(1^2 + 1^2 + 2^2 + 3^2 + ..... + n^2)^(n/2)
= (1 + n*(n + 1)*(2*n + 1)/6)^(n/2)
Now the coefficient of x is the coefficient of 1/x.
Hence, when I multiply the two series, we get
( (1 + x +....+ x^n)*(x^n + x^(n - 1) +.....+ n)/x^n )^n
Hence, the coefficient of x^n should be the square root of the constant term.
Since the denominator(inside the outermost power) contains x^n, we need the coefficient of x^n in the numerator which is
(1^2 + 1^2 + 2^2 + 3^2 + ..... + n^2)^(n/2)
= (1 + n*(n + 1)*(2*n + 1)/6)^(n/2)
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