Question

1)Find S12 for 1 + 6 + 11 + 16 +…
2)Find S22 for -6 + (-4) + (-2) + 0 …
thanks for your help!

Answer 1

For the first series, \(S_1=1\) and \(S_2=1+5=S_1+5=6\) (a common difference of 5 between successive terms). Next, \(S_3=6+5=S_2+5=S_1+2(5)=11\).

Continuing the pattern, you can establish that\[\begin{align*}S_n&=S_{n-1}+5\\&=S_{n-2}+(2)5\\&=\cdots\\&=S_2+(n-2)5&\\&=S_1+(n-1)5\end{align*}\]
Then \(S_{12}=S_1+(12-1)5=\cdots\). You can use the same reasoning to figure out the second problem.