SL 1.2—Arithmetic sequences and series

Use the common difference to model an arithmetic sequence.

An arithmetic sequence changes by the same additive amount each step: aₙ=a₁+(n−1)d, and a finite sum is n/2[2a₁+(n−1)d].

Worked example
For 4, 7, 10, the difference is 3, so a₆=4+5(3)=19.

Worked example
What stays constant? the difference, not the ratio.

Common boundary
Do not use a geometric formula when the change is additive.

Complete finite-sum and sigma example: for $4,7,10,\ldots$, $a_1=4$ and $d=3$, so $a_n=4+3(n-1)$. For six terms, $S_6=\frac{6}{2}[2(4)+5(3)]=69$. The same sum is $\sum_{k=1}^{6}(3k+1)=69$. Use $a_n$ for one term and $S_n$ or $\Sigma$ for a total.