Question

Hello Mates,
Please answer it..​

Answer 1

Answer:

0.505

Step-by-step explanation:

Given: (1 - 1/2²)(1 - 1/3²)(1 - 1/4²)(1 - 1/5²) ............. (1 - 1/85²)

= (1 - 1/4)(1 - 1/9)(1 - 1/16)(1 - 1/25) ...... (1 - 1/7225)

= (3/4)(8/9)(15/16)(24/25).....(7224/7225)

Now,

(i)

(3/4) * (8/9)

= 24/36

= 6/9

(ii)

(3/4) * (8/9) * (15/16)

= (6/9) * (15/16)

= 90/96

= 10/16

(iii)

(3/4) * (8/9) * (15/16) * (24/25)

= (10/16) * (24/25)

= (240/400)

= 15/25

Here,

Numerator = 6,10,15... = (n²/2) + (3n/2) + 1 {Where n > 2}

Denominator = 9,16,25.. = (n + 1)² {Where n > 2}

Now,

Total number of terms n = 84.

⇒ (n²/2 + 3n/2 + 1)/(n + 1)²

⇒ (84²/2 + 126 + 1)/(85)²

⇒ ~0.505

Hope it helps!

Answer 2

1-1/k2=(k2-1)/k2=(k+1)(k-1)/k2

donc [2;2004] [1-1/k2]

= [2;2004] (k+1) [2;2004](k-1)/[ [2;2004]k [2;2004]k]

or

[2;2004](k+1)=\frac{1}{2}(2005!)

[2;2004](k-1)=2003!

[2;2004]k=2004!

et donc en remplaçant :

x=(1-1/2²)(1-1/3²)(1-1/4²)...(1-1/2004²)

= \frac{1}{2}(2005!)(2003!)/[(2004!)(2004!)]

=\frac{1}{2}[2005!/2004!][2003!/2004!]