The sum of the series 1² 1 2 + 1²+2² 2 3 + 1²+2²+3² 3 4 + upto 20 terms is

Step-by-step solution

Let S= 1² 1 2 + 1²+2² 2 3 + 1²+2²+3² 3 4 + . upto 20 terms Let t_ n be n th terms of series. Then, t_ n = 1²+2²+3²+ +n² n (n+1) = n² n(n+1) = n(n+1)(2 n+1) 6 n(n+1) = 2 n+1 6 Taking summation on both sides aligned _ n=1 ²⁰ t_ n &= 2 6 _ n=1 ²⁰ n+ 1 6 _ n=1 ²⁰ 1 &= 1 3 ( 20(20+1) 2 )+ 1 6 20 &= 1 3 ( 20 21 2 )+ ( 10 3 )= 210 3 + 10 3 = 220 3 aligned