Value of (1/2)(1-1/3).......(1-1/n-1)(1-1/n)

Open in App

Solution

(1/2)(1-1/3).......(1-1/n-1)(1-1/n)

I can write this as
( 1/2 )( 2/3 )( 3/4 )( 4/5 ) . . . ( n-2)/( n-1 )( n-1 )/( n )
There are cancellations on the top and bottom
I can rewrite the whole thing as
(2/2)(3/3)4/4) . . . (n-1)/(n-1) * (1/n)

This equals 1/n
---------------
You can test it for n = 5, for instance
( 1/2)(2/3)(3/4)(4/5) =
(2/2)(3/3)(4/4)(1/5) = 1/5

Value of (1/2)(1-1/3).......(1-1/n-1)(1-1/n) = 1/n

Suggest Corrections

Similar questions

Prove $(1 + \frac{1}{1}) (1 + \frac{1}{2}) (1 + \frac{1}{3}) \dots (1 + \frac{1}{n}) = (n + 1) .$

(1 + \frac{1}{1}) (1 + \frac{1}{2}) (1 + \frac{1}{3}) . . . (1 + \frac{1}{n}) = (n + 1) .

$(1 + \frac{1}{1}) (1 + \frac{1}{2}) (1 + \frac{1}{3}) . . . . . . . . . . . . . . . . . . (1 + \frac{1}{n}) =$

\frac{3}{1 \cdot 2} (\frac{1}{2}) + \frac{4}{2 \cdot 3} {(\frac{1}{2})}^{2} + \frac{5}{3 \cdot 4} {(\frac{1}{2})}^{3} + \dots

n

Join BYJU'S Learning Program