Question

The sum of 3/1.2 . 1/2 + 4/2.3(1/2)² + 5/3.4(1/2)³ + .....n terms is

(a)  1 - 1(n + 1)2ⁿ

(b) 1 - 1/n.2^{n - 1}

(c)  1 + 1/(n + 1)2ⁿ

(d) None of these

Answer 1

Correct option  (a)  1 - 1(n + 1)2ⁿ 

Explanation:

T_n = n + 2/n.(n + 1) . 1/2ⁿ = (2/n - 1/n + 1). 1/2ⁿ

T_n =  1/n. 2^{n - 1} - 1/(n + 1). 2ⁿ

Putting n = 1, 2, 3, ..............,n

T_n =  1/n. 2^{n - 1} - 1/(n + 1). 2ⁿ

Adding , T₁ + T₂ + T₃ +.........+ T_n = S_n

S_n = 1 - 1(n + 1)2ⁿ