Question

The arithmetic mean of the co-efficients in the expansion of $(1+x{)}^{30}$ is

• A. $\frac{2^{30}}{31}$
• B. $\frac{2^{30}+1}{31}$
• C. $\frac{2^{31}}{31}$
• D. $\frac{2^{30}}{30}$

Answer 1

The correct option is A $\frac{2^{30}}{31}$

$(1+x{)}^{30}=C_0+C_{1}x+C_2{x}^2+\cdots +C_{30}{x}^{30}$
Putting $x=1$ on both sides
$2^{30}=C_0+C_1+C_2+\cdots +C_{30}\cdots \left(1\right)$
Number of terms $=30+1=31$
Now,
Arithmetic Mean$=\frac{C_0+C_1+C_2+\cdots +C_{30}}{31}$
$\therefore$Arithmetc Mean $=\frac{2^{30}}{31}$ from $\left(1\right)$