Algebra-11 (CAT-2000)

What is the value of the following expression:

(1/(2²-1))+ (1/(4²-1))+ (1/(6²-1))+……….. (1/(20²-1))

(1)9/19      (2) 10/19        (3) 10/21        (4) 11/21

Solution follows here:

Solution:

This is a problem based on algebraic simplification and manipulation.

(1/(2²-1))+ (1/(4²-1))+ (1/(6²-1))+……….. (1/(20²-1))

Using the formula a²-b² = (a+b)(a-b),

= (1/(2-1)(2+1))+ (1/(4-1)(4+1))+ (1/(6-1)(6+1))+…………….. (1/(20-1)(20+1))

= 1/1*3 + 1/3*5 + 1/5*7 + .. 1/19*21

Multiplying and dividing with ‘2’,

= ½(2/1*3 + 2/3*5 + 2/5*7 + .. 2/19*21)

= ½((3-1)/1*3 + (5-3)/3*5 + (7-5)/5*7 + .. (21-19)/19*21)

= ½((1/1 – 1/3) + (1/3 – 1/5) + (1/5 – 1/7) + …….(1/19 – 1/21)

= ½ (1/1 – 1/3 + 1/3 – 1/5 + 1/5 – 1/7 + ……..+1/19 – 1/21)

= ½ (1 – 1/21) = ½ (20/21) = 10/21    

Answer (3)