Question 1194991
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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[n]}}}.



Notice (and it is OBVIOUS) that  {{{a[n]}}} = {{{S[n]}}} - {{{S[n-1]}}}.


Substitute here the formulas for  {{{S[n]}}}  and  {{{S[m-1]}}}

    {{{a[n]}}} = {{{S[n]}}}- {{{S[n-1]}}} = {{{(n^2+2n)}}} - {{{((n-1)^2+2*(n-1))}}}


and simplify it

               = {{{n^2 + 2n - (n^2 -2n + 1 +2n -2)}}} = 2n + 1.


Thus the formula for the n-th term is

    {{{a[n]}}} = 2n + 1,  n = 1, 2, 3, 4, 5, . . . 


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

    {{{a[1]}}} = 3;   {{{a[2]}}} = 5;  {{{a[3]}}} = 7.
</pre>

Solved and clearly explained.


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