Chapter 12.5, Problem 10E

To determine

To Prove: $1^3+3^3+5^3+\mathrm{.....}+{\left(2n-1\right)}^3=n^2\left(2n^2-1\right)$ is True for all natural numbers $n$ .

Explanation of Solution

Given information:

Use Mathematical induction to prove the above mentioned formula.

Concept used:

1.By using Mathematical induction technique we need to prove that this formula holds True for all natural numbers. The first step is to prove True for $n=1$ .

2.Second step by using induction hypothesis of mathematical induction we assume for $n=k$ given formula is True.

3.Third step is we need to prove that above formula is True for $n=k+1$ by using formula is True for $n=k$ .

Proof:

First, we need to prove the above formula is True for $n=1$ .

Given formula is

  $1^3+3^3+5^3+\mathrm{.....}+{\left(2n-1\right)}^3=n^2\left(2n^2-1\right)$

Put $n=1$

We get

Left hand side = $1^3=1$

Right hand side = $1^2\left(2{\left(1\right)}^2-1\right)=2-1=1$

So Left hand side = Right hand side

It True for $n=1$ .

Let us assume it is True for $n=k$

So

  $1^3+3^3+5^3+\mathrm{.....}+{\left(2k-1\right)}^3=k^2\left(2k^2-1\right)$ is True

We need to prove it is True for $n=k+1$

Left hand side = $1^3+3^3+5^3+\mathrm{.....}+{\left(2k-1\right)}^3+{\left(2\left(k+1\right)-1\right)}^3$

  $\Rightarrow$ Left hand side = $k^2\left(2k^2-1\right)+{\left(2k+2-1\right)}^3$

  $\Rightarrow$ Left hand side = $k^2\left(2k^2-1\right)+{\left(2k+2-1\right)}^3$

  $\Rightarrow$ Left hand side = $2k^4-k^2+8k^3+1+12k^2+6k$

  $\Rightarrow$ Left hand side = $2k^4+8k^3+11k^2+6k+1$

Right hand side = ${\left(k+1\right)}^2\left(2{\left(k+1\right)}^2-1\right)$

  $\Rightarrow$ Right hand side = $\left(k^2+2k+1\right)\left(2\left(k^2+2k+1\right)-1\right)$

  $\Rightarrow$ Right hand side = $\left(k^2+2k+1\right)\left(2k^2+4k+2-1\right)$

  $\Rightarrow$ Right hand side = $\left(k^2+2k+1\right)\left(2k^2+4k+1\right)$

  $\Rightarrow$ Right hand side = $2k^4+4k^3+k^2+4k^3+8k^2+2k+2k^2+4k+1$

  $\Rightarrow$ Right hand side = $2k^4+8k^3+11k^2+6k+1$

Left hand side = Right hand side

And hence the proof by Mathematical induction.