Question

Given the series:

$2.5+{3.5}^2+{4.5}^3+........+(n+1)5^n$

($T_n$ and $S_n$ represent the $n^{th}$ term and sum of first $n^{th}$ term)

Calculate $T_{51}-T_{50}$?

• A. $1045\times 5^{49}$
• B. $210\times 5^{50}$
• C. $260\times 5^{50}$
• D. None of these

Answer 1

The correct option is A $1045\times 5^{49}$

Given Series:

$2.5+{3.5}^2+{4.5}^3+..........+(n+1)5^n$

To Find:

$T_{51}-T_{50}=?$

Solution:

$T_n=(n+1)5^n$

$T_{51}=(51+1)5^{51}={52.5}^{51}$

$T_{50}=(50+1)5^{50}={51.5}^{50}$

Now,

$T_{51}-T_{50}={52.5}^{51}-{51.5}^{50}=5^{50}(52\times 5-51)=5^{50}\times 209=1045\times 5^{49}$

Hence, A is the correct option.