The sum of the series 1/1*3 + 1/3*5 + 1/5*7 ... + 1/47*49 will be 24/49.
What is a sequence?
The sequence is the arrangement of numbers in a particular order.
What is a series?
A Series is the sum of elements that are arranged in a sequence.
Given here the series is
1/1*3 + 1/3*5 + 1/5*7 ... + 1/47*49
Taking the first term and separating each part as sum of two fractions
1/1*3= (1/2)(1/1 - 1/3)
1/3*5= (1/2)(1/3 - 1/5)
1/5*7= (1/2)(1/5 - 1/7)
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similarly
1/47*49= (1/2)(1/47 - 1/49)
Then series can be written as
1/1*3 + 1/3*5 + 1/5*7 ... + 1/47*49
= (1/2)(1/1 - 1/3) + (1/2)(1/3 - 1/5) + (1/2)(1/5 - 1/7) +...............+ (1/2)(1/47 - 1/49)
taking 1/2 as common
=(1/2)( 1- 1/3+1/3 -1/5+1/5 -1/7+1/7 .........+1/47-1/49)
canceling all the positive negative like terms
= (1/2)( 1- 1/49)
= (1/2)( 48/49)
= 24/49
Therefore the sum of the series 1/1*3 + 1/3*5 + 1/5*7 ... + 1/47*49 will be 24/49.
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