Question

Find the sum of the series $1.3+{2.3}^2+{3.3}^3+.......+{100.3}^{100}$

• A. $49.75\times 3^{100}+0.75$
• B. $99.5\times 3^{100}+1.5$
• C. $49.75\times 3^{100}-1.5$
• D. None of these

Answer 1

The correct option is D None of these

Given series:

$1.3+{2.3}^2+{3.3}^3+.........+{100.3}^{100}$

Solution:

Let,

$S_n=1.3+{2.3}^2+{3.3}^3+...........+{100.3}^{100}$ .............(i)

$3.S_n={1.3}^2+{2.3}^3+.............+{99.3}^{100}+{100.3}^{101}$ ..............(ii)

Subtracting eqn. (i) from (ii), we get

$-2.S_n=1.3+3^2+3^3+3^4+............+3^{100}-{100.3}^{101}$

or. $-2.S_n=\frac{3(3^{100}-1)}{(3-1)}-{100.3}^{101}$

or, $S_n=50\times 3^{101}-\frac{3(3^{100}-1)}{4}$

or, $S_n=50\times 3^{101}-0.25\times 3^{101}+0.75$

or, $S_n=49.75\times 3^{101}+0.75$

Hence, D is the correct option.