Correct Answer - Option 3 : 99×2
100 + 1
Concept:
If a be the first term, r be the common ratio of a GP then, \(\rm S_n = \frac{a(1-r^n)}{1-r}=\frac{a}{1-r}-\frac{ar^{n}}{1-r}\)
Let S∞ denote the sum of the infinite term of GP, then \(\rm S{_{∞}}= \frac{a}{1-r}\),
where -1 < r < 1.
Here a = First term of G.P and r = Common ratio.
An Arithmetic-Geometric Progression (AGP) is a progression in which each term can be represented as the product of the terms of arithmetic progressions (AP) and geometric progressions (GP).
Calculation:
Let,
S = 1 + 2×2 + 3×22 + 4×23 + ..... + 100×299
Above series is AGP.
⇒ 2S = 1×2 + 2×22 + 3×23 + ....+ 100×2100
⇒ S - 2S = (20 + 21 + 22 + ...299 ) - 100×2100
⇒ 100.2100 - S = \( = \frac{1.(2^{100}\ -\ 1)}{2\ -\ 1}\)
⇒ S = 99×2100 + 1