Question

In Problem, prove each statement.

$a+b$is a factor of$a^{2n+1}+b^{2n+1}$

Short answer

The given statement is shown.

Step-by-step solution

Step 1. Given information

The given expression is$a+b$.

Step 2. Prove the given statement.

First we consider the statement for $n=1$.

$\begin{array}{rcl}{a}^{2.1+1}+b^{2.1+1}& =& a^3+b^3\\ & =& (a+b)\left(a^2+b^2-ab\right)\end{array}$

which is divisible by $a+b$

Hence, condition holds true.

Now we assume the statement is true for some $k$

Now we consider,

$\begin{array}{rcl}{a}^{2(k+1)+1}+b^{2(k+1)+1}& =& a^{2k+3}+b^{2k+3}\\ & =& a^{2k+1}·a^2+b^{2k+1}·b^2\\ & =& \left(a^{2k+1}+b^{2k+1}-b^{2k+1}\right)·a^2+b^{2k+1}{b}^2\\ & =& \left(a^{2k+1}+b^{2k+1}\right)a^2-a^2{b}^{2k+1}+b^{2k+1}{b}^2\\ & =& \left(a^{2k+1}+b^{2k+1}\right)a^2+b^{2k+1}\left(b^2-a^2\right)\end{array}$

Since $\left(a^{2k+1}+b^{2k+1}\right)$is divisible by $a+b$and $b^{2k+1}\left(b^2-a^2\right)$is also divisible by $a+b$, we say that condition also holds true.