Answer
$S_{1}=1$
$S_{2}=4$
$S_{3}=9$
$S_{4}=16$
$S_{5}=25$
$S_{6}=36$
Work Step by Step
We are given:
$1, 3, 5, 7$ ...
We notice that the terms go up by $2$. We find the pattern:
$a_1=1+2(1-1)=1$
$a_2=1+2(2-1)=3$
$a_3=1+2(3-1)=5$
$a_4=1+2(4-1)=7$
Therefore:
$a_n=1+2(n-1)$
$a_{n}=2n-1$
So:
$a_5=1+2(5-1)=9$
$a_6=1+2(6-1)=11$
We find the partial sums:
$S_{1}=1$
$S_{2}=1+3=4$
$S_{3}=1+3+5=9$
$S_{4}=1+3+5+7=16$
$S_{5}=1+3+5+7+9=25$
$S_{6}=1+3+5+7+9+11=36$
