A Show that . 2. b Copy and complete: aligned (0 + 1)^3 &= 0^3 + 3(0)^2 + 3(0) + 1 (1…

MATHEMATICS · Multi-part · ISC · Mathematics · Algebra · Algebra

Problem

1. a Show that

(x + 1)^3 = x^3 + 3x^2 + 3x + 1

. 2. b Copy and complete:

\begin{aligned} (0 + 1)^3 &= 0^3 + 3(0)^2 + 3(0) + 1 \ (1 + 1)^3 &= 1^3 + 3(1)^2 + 3(1) + 1 \ (2 + 1)^3 &= 2^3 + \dots \ (3 + 1)^3 &= \dots \ &\vdots \ (n + 1)^3 &= \dots \end{aligned}

(a) Show that

(x + 1)^3 = x^3 + 3x^2 + 3x + 1

. (b) Copy and complete:

\begin{aligned} (0 + 1)^3 &= 0^3 + 3(0)^2 + 3(0) + 1 \ (1 + 1)^3 &= 1^3 + 3(1)^2 + 3(1) + 1 \ (2 + 1)^3 &= 2^3 + \dots \ (3 + 1)^3 &= \dots \ &\vdots \ (n + 1)^3 &= \dots \end{aligned}

(n + 1)^3 = 3 \sum_{k=1}^{n} k^2 + 3 \sum_{k=1}^{n} k + (n + 1)

. (d) Given that

\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}

, show that

\sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6}