Question

Find the value of ( 1 - 1/2² ) ( 1 - 1/3² ) ( 1 - 1/4² ) ...................................... ( 1 - 1/85² ).

Answer 1

Answer:

89 / 176

Step-by-step explanation:

I observed a pattern while reducing the number of terms.

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, $n^{th}$ 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 $n^{th}$ term is 2(n + 2) - 1 ]

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

= 7568

Answer = 3827 / 7568

             = 89 / 176

Answer 2

Answer:

43/85

Step-by-step explanation:

Find the value of    ( 1 - 1/2² ) ( 1 - 1/3² ) ( 1 - 1/4² ) ............... ( 1 - 1/85²)

================

Solution:

              ⇒  Use of formula: a²-b²=(a+b)(a-b)

We can write each term as:

• 1-1/2²= (1+1/2)(1-1/2)= 3/2*1/2

• 1-1/3²=(1+1/3)(1-1/3)= 4/3*2/3
• 1-1/4²=(1+1/4)(1-1/4)= 5/4*3/4

• ...

• 1-1/85²=(1+1/85)(1-1/85)= 86/85*84/85

Now replacing each term in the original expression:

•  3/2*1/2*4/3*2/3*5/4*3/4*.....*86/85*84/85  = 1/2*86/85=43/85

As we see all terms eliminated leaving simple fraction in the end

Answer is 43/85