Find the sum of the first 10 terms in the series 1 × 2, 2 × 3, 3 × 4, .... :
Read Explanation:
n th term of the sequence= n(n + 1)
Sum of first 10 terms of the sequence = 1 × 2 + ( 2 × 3) + ( 3 × 4 ) + .....+ (10×11)
= 1 × (1 + 1) + 2(2+1) + 3(3+1) + ........ + 10(10 + 1)
= 1² + 1 + 2² + 2 + 3² + 3 + 4² + 4 + ....... + 10² + 10
= 1² + 2² + 3² + ...... + 10² + 1 + 2 + 3 + ...... + 10
= Sum of squares of the first n numbers
+ sum of the first n numbers
= n(n+ 1)(2n+ 1)/6 + n(n+1)/2
= 10 × 11 × 21/6 + 10 × 11/2
= 385 + 55
= 440
