Question

if x=1/2*3+1/3*4+.....+1/63*64 then find the value of x​

Answer 1

Answer:

x = 31/64

Step-by-step explanation:

x=1/2*3+1/3*4+.....+1/63*64  , find x ?

Consider 1/1*2+1/2*3+1/3*4+.....+1/63*64

=> the general term is 1/n(n+1)

let us assume 1/n(n+1)  = A/n  + B/(n+1)    , where A & B are any arbitrary value

=>                    1/n(n+1)  = A(n+1) + B(n) / n(n+1)

=>                    1 = An+A + Bn

=>                    1 = A + n(A+B)

if we have to take A=1

=> A = 1  so A+B = 0                (as 'n' cannot be zero)

=> B = -1

and so the general term 1/n(n+1)  = 1/n  - 1/(n+1)

and hence  1/1*2+1/2*3+1/3*4+.....+1/63*64

                = (1/1 - 1/2)  + (1/2 - 1/3) + (1/3 - 1/4) ....... +(1/63 - 1/64)

                = 1 - 1/64    (on cancelling the -ve and +ve no.s)

                = 63/64

so  1/1*2+1/2*3+1/3*4+.....+1/63*64 = 1/1*2 + x = 63/64

                                                      => x = 63/64 - 1/2

                                                      => x = 126-64 / 128 = 62/128

                                                      => x = 31/64