Log in to add a comment

Log in to add a comment

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

Let us add 1 in the beginning for the series. We will subtract 1 from the sum of (N+1) terms to get the answer.

*Terms are : 1, 3, 7, 13, 21, 31, 43, 57, ..*

*Differences between terms are : d_n = 2, 4, 6, 8, 12, 14*

nth term of the series of differences = d_n = 2 * n.

*Sum of this series for n terms = 2 * n(n+1)/2 = n² + n*

So the series is :

T1 = 1

T_n+1 = T1 + n² + n , for n >= 1

*Rewrite it:*

T_n+1 = 1 + n + n² = (n+1)² - (n+1) + 1

=>* T_n = n² - n + 1 for n>= 1*

*So sum of the series for (n+1) terms is now: *

= Σ n² - Σ n + Σ 1

= (n+1) (n+2) (2 n + 3) / 6 - (n+1) (n+2)/2 + (n+1)

= (n +1) / 6) [ 2 n² + 7n + 6 - 3 n - 6 + 6 ]

= (n+1) (n² + 2 n + 3) / 3

= (n³ + 3 n² + 5 n + 3) / 3

*Sum of the given series for n terms is now : * (subtract 1)

= (n³ + 3 n² + 5 n + 3 ) / 3 - 1

=*(n³ + 3 n² + 5 n) / 3*

============================

__General procedure for such series.__

1) Find the series for the differences among terms.

2) Express the n th term in the form of n.

3) Find the sum of n terms of the series of differences.

4) Then add this to the first term T1 of the given series.

5) Now find the sum in terms of Σ n, Σn² etc.

nth term of the series of differences = d_n = 2 * n.

So the series is :

T1 = 1

T_n+1 = T1 + n² + n , for n >= 1

T_n+1 = 1 + n + n² = (n+1)² - (n+1) + 1

=>

= Σ n² - Σ n + Σ 1

= (n+1) (n+2) (2 n + 3) / 6 - (n+1) (n+2)/2 + (n+1)

= (n +1) / 6) [ 2 n² + 7n + 6 - 3 n - 6 + 6 ]

= (n+1) (n² + 2 n + 3) / 3

= (n³ + 3 n² + 5 n + 3) / 3

= (n³ + 3 n² + 5 n + 3 ) / 3 - 1

=

============================

1) Find the series for the differences among terms.

2) Express the n th term in the form of n.

3) Find the sum of n terms of the series of differences.

4) Then add this to the first term T1 of the given series.

5) Now find the sum in terms of Σ n, Σn² etc.