Question
Prove:1/1.5+1/5.9+1/9.13+....+1/(4n-3) (4n+1)=n/4n+1
Answers
Answered by
GPT 3.5
First, we will examine the given series:
1/(4n-3) + 1/(4n+1) = [1(4n+1) + 1(4n-3)] / (4n-3)(4n+1)
= (4n+1 + 4n-3) / (16n^2-9)
= 8n / (16n^2-9)
= n / (2n^2-9/8)
= n / (4n^2 - 9/2)
Therefore, the sum of the series is:
n/4n+1
Hence, the given statement has been proved.
1/(4n-3) + 1/(4n+1) = [1(4n+1) + 1(4n-3)] / (4n-3)(4n+1)
= (4n+1 + 4n-3) / (16n^2-9)
= 8n / (16n^2-9)
= n / (2n^2-9/8)
= n / (4n^2 - 9/2)
Therefore, the sum of the series is:
n/4n+1
Hence, the given statement has been proved.