What is the sum of first 20 terms of the following series ?

1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + ..........

Question

Question

This question was previously asked in

SSC CGL Tier 2 Quant Previous Paper (Held On: 8 Aug 2022)

Answer 1

Given:

1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + .......... + 20

Concept used:

The sum of the first n consecutive no. is n(n+1)/2.

The sum of the square of the first n consecutive no. is n(n + 1)(2n + 1)/6.

Calculations:

According to question, we have:

1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + ..........up to 20 terms

nth term = n(n + 1)

Sum = ∑ n (n + 1) = ∑ (n² + n)

Sum = ∑ n² + ∑ n

Sum = \(\frac{n(n +1)(2n +1)}{6}+\frac{n(n +1)}{2}\)

Sum = \(\frac{n(n +1)}{2}\Big[\frac{2n +1}{3}+1\Big]\)

Sum = \(\frac{n(n +1)}{2}\Big[\frac{2n +4}{3}\Big]\)= \(\frac{n(n +1)(2n + 4)}{6}\)

Substitute n = 20, we have

Sum = \(\frac{20(20 +1)(2\times 20+ 4)}{6}\)

⇒ Sum = \(\frac{20\times 21 \times 44}{6}\)

⇒ Sum = \(\frac{420 \times 44}{6}\)

⇒ Sum = 3080

∴ The sum of first 20 terms of the given series is 3080.