Question

find the value of (1- 1 by 2²) (1-1 by 3²) (1- 1 by 4²) (1 - 1 by 5²) ............(1-1 by 9²) ( 1- 1 by 10²)​

Answer 1

For 2 terms , i.e. 1-1/2² x 1-1/3², answer is 2/3

For 3 terms, answer = 5/8

For 4 terms, answer = 3/5 = 9/15

and so on.

We can see that the numerator increases by 3,4,5,6 ...

For denominators, the increase pattern is 5,7,9,11...

So answer = numerator of last term / denominator

Numerator will be 2 + (sum of no. from 3 to 87)

[observing the pattern, n^{th}n

th

term is n+2 ]

= 2 + sum of no. from 1 to 87 - sum of 1 and 2

= 2 + 87(44) - 3 [Using sum of 'n' natural no. is n(n+1) / 2]

= 3827

Similarly, denominator will be 3 + (5+7+9...)

= 3 + (1 + 3 + 5 + 7 ...... 87(2) - 1 ) - (1 + 3)

[There are 85 terms and the n^{th}n

th

term is 2(n + 2) - 1 ]

= 3 + 87² - 4 [ Sum of 'n' odd no. = n²]

= 7568

Answer = 3827 / 7568

= 89 / 176

HOPE IT HELPS YOU