Question

If $\sum _{r=6}^{n+5}(r-5{)}^3={\left[\frac{p(p-1)}{2}\right]}^2$ is true for $n\in N,$ then $p=$
(use principle of mathematical induction)

• A. $n+2$
• B. $n+1$
• C. $n-1$
• D. $n+3$

Answer 1

The correct option is B $n+1$

Using the principle of mathematical induction it can be proved that,
$1^3+2^3+3^3+\cdots +n^3={\left[\frac{n(n+1)}{2}\right]}^2$
$\therefore \sum _{r=6}^{n+5}(r-5{)}^3=1^3+2^3+3^3\cdots +n^3=\frac{n^2(n+1{)}^2}{4}={\left(\frac{n(n+1)}{2}\right)}^2={\left(\frac{(p-1)p}{2}\right)}^2$
$\therefore p=n+1$