Home
Class 12
MATHS
Sum of the n terms of the series 1 . 2^(...

Sum of the n terms of the series `1 . 2^(2)+2 . 3^(2)+3 . 4^(2)+..`

Text Solution

Verified by Experts

Let `T_(n)` be the nth rerm of this series, then
`T_(n)=(" nth term of "1,2,3,"...") (" nth term of "2^(2),3^(2),4^(2),".....")`
`=n(n+1)^(2)=n^(3)+2n+n`
` therefore " sum of n terms " S_(n)= sumT_(n)`
`2=sumn^(3)+2sumn^(2)+sumn`
`={(n(n+1))/(2)}+2{(n(n+1)(2n+1))/(6)}+(n(n+1))/(2)`
`=(n(n+1))/(2){(n(n+1))/(2)+(2(2n+1))/(3)+1}`
`=(n(n+1))/(12)(3n^(2)+3n+8n+4+6)`
`=(n(n+1)(3n^(2)+11n+10))/(12)=(n(n+1)(n+2)(3n+5))/(12)`
Promotional Banner

Similar Questions

Explore conceptually related problems

The sum of first n terms of the series 1^(2) + 2.2^(2) +3^(2) + 2. 4^(2) + 5^(2) + 2. 6^(2) + "........." is ( n ( n + 1)^(2))/( 2) when n is even. When, n is odd, the sum is :

Sum of 25 terms of the series 1+2 . 2+3 . 2^(2)+4 . 2^(3)+5 . 2^(4)+..

Find the sum to n terms of the series . 3 xx 1^(2) + 5 xx 2^(2) + 7 xx 3^(2) + . . . . . .

Find the sum of the n terms to the series 5^(2)+6^(2)+7^(2)+......+20^(2) ?

The sum of 1^(st) n terms of the series (1^(2))/(1) + (1^(2) + 2^(2))/(1 + 2) + (1^(2) + 2^(2) + 3^(2))/(1 + 2 + 3) + ..

Sum to n terms of the series 2+6+12+20+.. is,

The sum of n terms of the series 1^(2)-2^(2)+3^(2)-4^(2)+5^(2)-6^(2)+... is

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

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

Statement 1 The sum of first n terms of the series 1^(2)-2^(2)+3^(2)-4^(2)_5^(2)-"……" can be =+-(n(n+1))/(2) ... Statement 2 Sum of first n narural numbers is (n(n+1))/(2)