Question

The sum of n term of 1×3+3×5+5×7+.....

Answer 1

Step-by-step explanation:

The sum of n term of 1×3+3×5+5×Input : n = 2

Output : S_n = 18

Explanation:

The sum of first 2 terms of Series is

1*3 + 3*5

= 3 + 15

= 28

Input : n = 4

Output : S_n = 116

Explanation:

The sum of first 4 terms of Series is

1*3 + 3*5 + 5*7 + 7*9

= 3 + 15 + 35 + 63

= 116

Let, the n-th term be denoted by tn.

This problem can easily be solved by observing that the nth term can be founded by following method:

tn = (n-th term of (1, 3, 5, … ) )*(nth term of (3, 5, 7, ….))

Now, n-th term of series 1, 3, 5 is given by 2*n-1

and, the n-th term of series 3, 5, 7 is given by 2*n+1

Putting these two values in tn:

tn = (2*n-1)*(2*n+1) = 4*n*n-1

Now, the sum of first n terms will be given by :

Sn = ∑(4*n*n – 1)

=∑4*{n*n}-∑(1)

Now, it is known that the sum of first n terms of series n*n (1, 4, 9, …) is given by: n*(n+1)*(2*n+1)/6

And sum of n number of 1’s is n itself.

Now, putting values in Sn;

Sn=4*n*(n+1) *(2*n+1)/6-n

=n*(4*n*n+6*n-1) /3