Solution for Section 4.1.5

Solution for Section 4.1 Question 5

5. Two possible solutions are given here.

Solution 1: In the given equation, replace every occurence of n by (n + 1).

n + 1

i²

(n + 1) (n + 1 + 1) (2(n + 1) + 1)

i = 1

(n + 1) (n + 2) (2n + 3)

Solution 2: Split up the required sum into the sum of the first n terms and the final term. Then substitute and simplify.

n + 1

i²

(n + 1)²

i = 1

n (n + 1) (2n + 1)	+	6 (n + 1)²
6	+	6

(n + 1) [n (2n + 1) + 6 (n + 1)]
6

(n + 1) [ 2n² + 7n + 6]
6

(n + 1) (n + 2) (2n + 3)
6