SOLUTION: the sum of the first n terms of a series is given by Sn=n^2+2n .Find the rth term and the first three terms of the series.

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Question 1194991: the sum of the first n terms of a series is given by Sn=n^2+2n .Find the rth term and the first three terms of the series.
Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
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the sum of the first n terms of a series is given by Sn=n^2+2n .
Find the rth term and the first three terms of the series.
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I will explain you how to derive the formula for the n-th term  a%5Bn%5D.



Notice (and it is OBVIOUS) that  a%5Bn%5D = S%5Bn%5D - S%5Bn-1%5D.


Substitute here the formulas for  S%5Bn%5D  and  S%5Bm-1%5D

    a%5Bn%5D = S%5Bn%5D- S%5Bn-1%5D = %28n%5E2%2B2n%29 - %28%28n-1%29%5E2%2B2%2A%28n-1%29%29


and simplify it

               = n%5E2+%2B+2n+-+%28n%5E2+-2n+%2B+1+%2B2n+-2%29 = 2n + 1.


Thus the formula for the n-th term is

    a%5Bn%5D = 2n + 1,  n = 1, 2, 3, 4, 5, . . . 


To get first 3 term, substitute n= 1, 2, 3 into this formula and calculate

    a%5B1%5D = 3;   a%5B2%5D = 5;  a%5B3%5D = 7.

Solved and clearly explained.

Let me know if any point of my explanation is clear to you . . .