Question

In Un sequence , if U1=1/4 and Un+1=Un/2+Un then 1/U50 =

Answer 1

Given:

In Un sequence , U1 = 1/4 and Un+1 = Un / 2+Un

To find:

1/U50 = ???

Solution:

From given, we have,

U1 = 1/4 and

Un+1 = Un / 2+Un

⇒ 1/Un+1 = 2/Un + 1

Put n = 1

1/U2 = 2/U1 + 1

Put n = 2

1/U3 = 2/U2 + 1 = 2 (2/U1 + 1) + 1 = 4/U1 + 3 = 2²/U1 + (2² - 1)

Put n = 3

1/U4 = 2/U3 + 1 = 2³/U1 + (2³ - 1)

As we need to find 1/U50, we get,

1/U50 = 2^(50-1)/U1 + (2^(49) - 1)

= 2^(49)/U1 + (2^(49) - 1)

= 4 × 2^(49) + (2^(49) - 1)          [ ∵ U1 = 1/4 ]

= 5 × 2^(49)  - 1

∴ 1/U50 = 5×2^(49) - 1