Question

Prove by method of induction, for all $n\in$
$3+7+11+...$ to $n$ terms $=n(2n+1)$

State True or False

• A. True
• B. False

Answer 1

The correct option is A True

Given statement $P\left(n\right)$:

$3+7+11+....$ to $n$ terms $=n(2n+1)$

$\begin{array}{c}Putn=1\\ L.HS=3\\ R.H.S=1.\left(2.1+1\right)=3\\ L.HS=R.HS\\ P\left(1\right)istrue.\end{array}$

Assuming $P\left(k\right)$ is true, where $k\in N$

$P\left(k\right):3+7+.....$ to $k$ terms $=k(2k+1)...............\left(1\right)$

Consider $P(k+1)$

$L.H.S=3+7+............$ to $k+1$ term

$=3+7+.......$ to $k$ terms +$(4k+3)$

Where $4k+3$ is $(k+1)$th term of given series.

Using in $L.H.S$ we get

$\begin{array}{c}L.H.S=k\left(2k+1\right)+4k+3\\ =2k^2+5k+3\\ =\left(k+1\right)\left(2k+3\right)\\ =\left(k+1\right)\left[2\left(k+1\right)+1\right]\end{array}$

Hence,

$P(k+1)$ is true.