The question asks to prove the relationship stated below via proof by induction. The general approach is to start with an n value of 1; validate whether the left-handed side and the right-handed side are equivalent. Then, entering induction: first an assumption is made for the value n=a. For n=a, the statement is assumed to be true; and when the statement is true for n=a, it must be true for n=a+1 also.
Here's the overall approach I made so far:
Yet, in the induction part for n=a+1, it seems that the desired result equivalent to the right-handed side does not get derived.
I want to ask whether an appropriate approach is made for this proof; how to organize the expression below in order to show the statement is valid.
Thanks.
Best Answer
Note that your induction hypothesis involves an an inequality and not an equation. You have to fix that.
To conclude, you can use the following:
\begin{align} \frac{a^k(a+1)+2(a+1)^k}{2} &= \frac{a(a)^{k-1}(a+1) + 2(a+1)^k}{2} \\&\le \frac{a(a+1)^{k-1}(a+1) + 2(a+1)^k}{2} \\&= \frac{(a+1)^k (a)+2(a+1)^k}{2} \\&= \frac{(a+1)^k(a+2)}{2} \end{align}