Home
Class 12
MATHS
Find the sum of the series 1*n+2*(n-1)+3...

Find the sum of the series `1*n+2*(n-1)+3*(n-2)+4*(n-3)+"..."+(n-1)*2+,*1` also, find the coefficient of `x^(n-1)` in th cxpansion of `(1+2x+3x^(2)+"...."nx^(n-1))^(2)`.

Text Solution

Verified by Experts

The rth term of the given series is
`T_(r)=r*(n-r+1)=(n+1)r-r^(2)`
`therefore` Sum of the series
`S_(n)=sum _(r=1)^(n)T_(r)=(n+1)sum _(r=1)^(n)r-sum _(r=1)^(n)r^(2)=(n+1)sumn-sumn^(2)`
`=(n+1)(n(n+1))/(2)-(n(n+1)(2n+1))/(6)`
`=(n(n+1))/(6)(3n+3-2n-1)=(n(n+1)(n+2))/(6)`
Now,
`(1+2x+3x^(2)+"..."+nx^(n-1))^(2)=(1+2x+3x^(2)+"..."+nx^(n-1))xx(1+2x+3x^(2)+"..."+nx^(n-1))`
`therefore " Coeffocient of "x^(n-1) " in "(1+2x+3x^(2)+"..."+nx^(n-1))^(2)`
`1*n+2*(n-1)+3*(n-2)+"..."+n*1`
`S_(n)=(n(n+1)(n+2))/(6)`
Promotional Banner

Similar Questions

Explore conceptually related problems

The coefficients of x^(n) in the expansion of (1+x)^(2n) and (1+x)^(2n-1) are in the ratio

Find the sum of n terms of the series 1. 2. 3+2. 3. 4+3. 4. 5+

If C_(n) is the coefficient of x^(n) in the expansion of (1+x)^(n) then C_(1)+2. C_(2)+3. C_(3)+..+n. C_(n)

Find the sum of the series 1 + (1 + x) + (1 + x + x^2) + ... to n terms, x≠ 1

The coefficients of x^(-n) in (1+x)^(n)(1+(1)/(x))^(n) is

Find the (n+1)th term from the end in the expansion of (2x - (1)/(x))^(3n)

Find the sum of the series (1^(2)+1)1!+(2^(2)+1)2!+(3^(2)+1)3!+ . .+(n^(2)+1)n! .

Sum to n terms of the series 1+(1+2)+(1+2+3)+.. ..

If A and B are coefficients of x^(n) in the expansion of (1+x)^(2n) and (1+x)^(2n-1) respectively, then A/B equals:

Prove that the coefficient of x^n in the expansion of (1+x)^(2n) is twice the coefficient of x^n in the expansion of (1+x)^(2n-1) .