Question

The sum of n term of a series is n² +2n for all values of n find the first three terms of series

Answer 1

We know that the sum of n terms of a series is:

$S=n^2+2n$

We have to find the 3 first terms of the series.

The first term will be equal to the sum of all the terms when n = 1, as it is the only term present. Then:

$a_1=S(1)=1^2+2\cdot1=1+2=3$

We can calculate the second term (a2) as:

$\begin{gathered} a_1+a_2=S(2) \\ a_2=S(2)-a_1 \\ a_2=(2^2+2\cdot2)-3 \\ a_2=(4+4)-3 \\ a_2=8-3 \\ a_2=5 \end{gathered}$

Finally, the third term will be:

$\begin{gathered} a_1+a_2+a_3=S(3) \\ a_3=S(3)-(a_1+a_2) \\ a_3=S(3)-S(2) \\ a_3=3^2+2\cdot3-2^2-2\cdot2 \\ a_3=9+6-4-4 \\ a_3=7 \end{gathered}$

NOTE: we can calculate any term (an) as:

$a_n=S(n)-S(n-1)$

Answer: the first 3 terms are a1 = 3, a2 = 5 and a3 = 7.