Question

Find the value of ( 1 - 1/2² ) ( 1 - 1/3² ) ( 1 - 1/4² ) ...................................... ( 1 - 1/2015²)(1-1/2016²)

Answer 1

Answer:

2017/4032

Step-by-step explanation:

Find the value of

( 1 - 1/2² ) ( 1 - 1/3² ) ( 1 - 1/4² ) ............... ( 1 - 1/2015²)(1-1/2016²)

Solution:

⇒  Use of formula: a²-b²=(a+b)(a-b)

We can change 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
• ...
• 1-1/2015²=(1+1/2015)(1-1/2015)= 2016/2015*2014/2015
• 1-1/2016²=(1+1/2016)(1-1/2016)= 2017/2016*2015/2016

Now replacing each term in the original expression:

• 3/2*1/2*4/3*2/3*5/4*3/4*.....*2016/2015*2014/2015*2017/2016*2015/2016= 1/2*2017/2016=2017/4032

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

Answer is 2017/4032