Question

Let A_n be the sum of the first n terms of the geometric series `704 + 704/2 + 704/4 + 704/8 + ...` and B_n be the sum of the first n terms of the geometric series `1984 - 1984/2 + 1984/4 + 1984/8 + ...` If A_n = B_n, then the value ofn is (where n ∈ N).

विकल्प

• 4

• 5

• 6

• 7

Answer 1

5

Explanation:

A_n = `704 + 704/2 + 704/4 + ...` to n terms

= `(704(1 - (1/2)^n))/(1 - 1/2) = 704 xx 2(1 - (1/2)^n)`

B_n = `1984 - 1984/2 + 1984/4 ....` to n terms

= `(1984(1 - ((-1)/2)^n))/(1 - ((-1)/2)) = 1984 xx 2/3(1 - ((-1)/2)^n)`

Now, A_n = B_n

⇒ `704 xx 2(1 - (1/2)^n)`

= `1984 xx 2/3 xx (1 - ((-1)/2)^n)`

⇒ 33 – 31 = `33(1/2)^n - 31((-1)/2)^n`

⇒ `2^(n + 1)` = 33 – 31(– 1)ⁿ 

⇒ n = 5