maths

13(a)

Σ r = (n/2)[2(1) + (n - 1)(1)] = n(n + 1)/2

(b) a^3 - (a - 1)^3

= a^3 - (a^3 - 3a^2 + 3a - 1)

= 3a^2 - 3a + 1

(c) Σ r^3 - (r - 1)^3

= Σ (3r^2 - 3r + 1)

= 3Σr^2 - 3n(n + 1)/2 + n

= 3Σr^2 - (3n^2 + n)/2

(d) Σ r^3 - (r - 1)^3

= (1^3 - 0^3) + (2^3 - 1^3) + ... + (n^3 - (n - 1)^3)

= n^3

3Σr^2 - (3n^2 + n)/2 = n^3

3Σr^2 = n^3 + (3n^2 + n)/2

Σr^2 = (2n^3 + 3n^2 + n)/6

Σr^2 = n(n + 1)(2n + 1)/6