Question

State True or False. The A.M of the series $1,4,16,....,4^n$ is $\frac{4^{n+1}-1}{3(n+1)}$

• A. True
• B. False

Answer 1

The correct option is A True

Consider the given series.

$1,4,16,64,......,4^n$

$A.M.=\frac{\text{Sum of all the numbers}}{\text{total numbers}}$

$A.M=\frac{1+4+16+.......+4^n}{n+1}$

$A.M=\frac{4^0+4^1+4^2+4^3+4^4+.......+4^n}{n+1}$

$4^0+4^1+4^2+4^3+4^4+.......+4^n$the given term is in G.P.

So sum is given by,

$S_n=\frac{a(r^n-1)}{r-1}$

$a=$ First term

$r=$ common ratio

$n=$ number of terms

$\Rightarrow$ $A.M=\frac{\frac{1(4^{n+1}-1)}{4-1}}{n+1}$

$A.M=\frac{4^{n+1}-1}{3(n+1)}$

Hence, this is the answer.