Question

$1.2+{2.2}^2+{3.2}^3+.................+n{.2}^n=(n-1)2^{n+1}+2$ is true for

• A. Only natural number $n$ $\ge$ $3$
• B. All natural number $n$
• C. Only natural number $n$ $\ge$ $5$
• D. None

Answer 1

The correct option is B All natural number $n$

Let P(n) be the given statement i.e. $P\left(n\right):1.2+{2.2}^2+{3.2}^3+.........+n{.2}^n=(n-1)2^{n+1}+2$
Putting n $=$ 1, LHS $=$ 1.2 $=$ 2; RHS $=$ 0+2 $=$ 2
$\therefore$ P(n) is true for $n=1$
Assume that P(n) is true for $n=k$
i.e. P(k) is true i.e. ${1.2}^2+{2.2}^2+{3.2}^3+...........+k.2$ ${}^k$ Replacing k by k + 1, we get the next term $=(k+1)2^{k+1}$
Adding it to both sides
$LHS=1.2+{2.2}^2+{3.2}^3+.........+k{.2}^k+(k+1)2^{k+1}$
$RHS=(k-1)2^{k+1}+2+(k+1)2^{k+1}$

$=2^{k+1}[k-1+k+1]+2=2k{2}^{k+1}+2=(k+1-1)2^{k+1+1}+2$

This proves P(n) true for $n=k+1$
Thus P(k+1) is true whenever P(k) is true
Hence. P(n) is true for all n $\in$ N