the value of 1/2^2-1+1/4^2-1+............+1/20^2-1
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Answer:
What is the value of [1 - 1/(2^2)] * [1 - 1/(3^2)] * [1 - 1/(4^2)] … [1 - 1/ (2017^2)]?
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Assume ak=1−1k2
=k2−1k2
=(k−1)(k+1)k2
Let Pn=∏k=2nan
So, P3=a2a3=322(2×432)
=42×3
Now, P4=P3a4
=42×3(3×542)
=52×4
So, we can prove by mathematical induction that
Pn=n+12n∀n≥2 where n∈N
So, P2017=20182×2017
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How do I evaluate (1−122)(1−132)...(1−120062)(1−120072) ?
What is the solution to (1-1/4) * (1-1/9) (1-1/16) (1-1/900)?
What is the value of (1/3) ^2?
How do I solve 1/12+1/22+1/32+…+1/20172<2 ?
What is 112+122+132+142+…?
1–1/(2²) is 1/1² - 1/2²
Which is (2²-1²)/2², which simplifies to,
(3)(1)/(2²)
Next term gives, (4)(2)/(3²)
The next is (3)(5)/4²
So, final product = 1×2×3²×4²….2016²×2017×2018/[2²×3²….2017²]
Multiplying numerator and denominator by 2,
We get, (2016!)² × (2017×2018)/2×(2017!)²
= 2018/2×2017
= 1009/2017 = 0.50024…..
Thank you!
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=>.[{1}^2-{1/2}^2]×[{1}^2-{1/3}^2]×[{1}^2-{1/4}^2]…………[{1}^2-{1/2016}^2]×[{1}^2-{1/2017}^2
=>(1–1/2)(1+1/2)×(1–1/3)(1+1/3)×(1–1/4)(1+1/4)×……..×(1–1/2016)(1+1/2016)×(1–1/2017)(1+1/2017).
=> 1/2×3/2×2/3×4/3×3/4×5/4××××××××××××2015/2016×2017/2016×2016/2017×2018/2017
=> 1/2×2018/2017 [All the fractions cancell each -other except first and last fraction.]
=> 1009/2017 answer.
It is a very simple solution.
You just need to write the first step, go on a push and the answer emerges.
I have attached an image. The solution procedure is also available on my youtube video channel.
How do I evaluate (1−122)(1−132)...(1−120062)(1−120072) ?
What is the solution to (1-1/4) * (1-1/9) (1-1/16) (1-1/900)?
What is the value of (1/3) ^2?
How do I solve 1/12+1/22+1/32+…+1/20172<2 ?
What is 112+122+132+142+…?
What is the value of (1-1/9) (1-1/16) (1-1/25) … (1-1/10000) is?
What is the value of {(-5)^2} ^1/2?
What is the value of (2 (2^2 (2^3 (2^4 (…) ^1/2) 1/2) 1/2) 1/2…) 1/2?