Question

In an Infinite Geometric progression, 20th term is 3a and sum of all subsequent terms is 7a. Which term of the progression is 3000a/343? ​

Answer 1

Answer:

If a is the first term, and r the common ratio:

Sum of an infinite GP = a/(1-r)

Squares of its terms: a^2, a^2r^2, a^2r^4,… is also an infinite GP with first term = a^2 and common ratio = r^2.

So, sum of squares of terms of original GP = a^2/(1-r^2)

Given that both are equal to 3:

a/(1-r) = 3 .. (1)

And a^2/(1-r^2) = 3

i.e a^2/((1-r)(1+r)) = 3 (using a^2 - b^2 = (a+b)(a-b))

Substituting (1) in above equation,

3*(a/(1+r)) = 3

i.e a/(1+r) = 1

i.e. a = 1+r .. (2)

Substituting (2) in (1),

(1+r)/(1-r) = 3

i.e. 1+r = 3–3r

i.e 4r = 2

So, r = 0.5

Substituting this in (2),

a = 1.5

So, first term is 1.5 and common ratio is 0.5.

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