Question

If $(1.5{)}^{30}=k$ then the value of $\sum _{n=2}^{29}(1.5{)}^n$ is?

• A. $2k-3$
• B. $k+1$
• C. $2k+7$
• D. None of these

Answer 1

The correct option is A $2k-3$

${\sum }_{n=2}^{29}{\left(1.5\right)}^n=S{\sum }_{n=1}^{30}{\left(1.5\right)}^n=1.5+{1.5}^2+{1.5}^3+...++{1.5}^{29}+{1.5}^{30}$

These are in GP $S=\frac{a(r^n-1)}{r-1}$

${\sum }_{n=1}^{30}{\left(1.5\right)}^n=\left(1.5\right)\left[\frac{{\left(1.5\right)}^{30}-1}{1.5-1}\right]=3({\left(1.5\right)}^{30}-1)=3(k-1)3(k-1)=1.5+{1.5}^2+{1.5}^3+...++{1.5}^{29}+{1.5}^{30}3(k-1)=1.5+{\sum }_{n=2}^{29}{\left(1.5\right)}^n+{1.5}^{30}3(k-1)=1.5+k+{\sum }_{n=2}^{29}{\left(1.5\right)}^{n}2k-4.5={\sum }_{n=2}^{29}{\left(1.5\right)}^n=S$