Question

1/1.3.5+1/3.5.7+......find sum of n term

Answer 1

1/1.3.5+1/3.5.7+......find sum of n term

=1/1.3.5+1/3.5.7+1/5.7.9 + ..................+1/(2n-1)(2n+1)(2n+3)

Now each term can be broken down as

1/1.3.5 = 1/4 [1/1.3 - 1/3.5]

1/3.5.7 = 1/4 [1/3.5 - 1/5.7]
...... ..............
...... ..............

1/(2n-1)(2n+1)(2n+3) = 1/4[1/(2n-1)(2n+1) - 1/(2n+1)(2n+3)]

Now putting the value we get,

1/4[1/1.3 - 1/3.5 +1/3.5 - 1/5.7+ 1/5.7-1/7.9+...........+1/(2n-1)(2n+1)-1/(2n+1)(2n+3)]

Now we see that terms will get cancelled and first and last term will be left i.e.

= 1/4[1/3 - 1/(2n+1)(2n+3)] Ans.