Use mathematical induction ,1/5.7 +1/7.9 +1/9.11 +........... upto n terms is equal to
please please solve this problem it's urgent
Answers
Given : 1/5.7 +1/7.9 +1/9.11 +........... upto n terms
To find : Value
Solution:
1/(5.7) + 1/(7.9) + 1/(9.11) + + 1/(2n+3)(2n+5)
= (1/2) { 2/(5.7) + 2/(7.9) + 2/(9.11) + + 2/(2n+3)(2n+5)}
= (1/2) {1/5 - 1/7 + 1/7 - 1/9 +1/9 - 1/11 + -1.(2n+3) +1/(2n +3) - 1/(2n + 5)}
= (1/2) { 1/5 - 1/(2n + 5)}
= (1/2) { 2n + 5 - 5} / 5(2n + 5)
= n /5(2n+5)
if using mathematical induction :
1/(5.7) + 1/(7.9) + 1/(9.11) + + 1/(2n+3)(2n+5) = n /5(2n+5)
n = 1
1/5.7 = 1/35
n /5(2n+5) = 1/(5)(7) = 1/35
let say n = k
=> 1/(5.7) + 1/(7.9) + 1/(9.11) + + 1/(2k+3)(2k+5) =k/5(2k+5)
to be proved n = k + 1
=> 1/(5.7) + 1/(7.9) + 1/(9.11) + + 1/(2k+5)(2k+7) = (k+1)/5(2k+7)
LHS = k/5(2k+5) + 1/(2k+5)(2k+7)
= 1/(2k+5) { k/5 + 1/(2k + 7)
= 1/(2k + 5) { 2k² + 7k + 5 }/5(2k + 7)
= 1/(2k + 5) { 2k² + 2k + 5k + 5 }/5(2k + 7)
= 1/(2k + 5) { (k + 1)(2k + 5) }/5(2k + 7)
= (k+1)/5(2k+7)
= RHS
QED
Hence proved
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