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1 + 2×2 + 3×22 + 4×23 + ..... + 100×299  is
1. 99×2100
2. 100×2100
3. 99×2100 + 1
4. 1000×2100

1 Answer

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Best answer
Correct Answer - Option 3 : 99×2100 + 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 + 4×2+ ..... + 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

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