Sequences and Series · Strategy summary

Sequences and Series: strategy summary

1. Arithmetic: add $d$ each time; $a_n = a_1 + (n - 1)d$. Geometric: multiply by $r$ each time; $a_n = a_1 r^{n-1}$.
2. From term 1 to term $n$ is $n - 1$ steps. From term $j$ to term $k$ is $k - j$ steps.
3. Two arithmetic terms: $d = \frac{\text{change in value}}{\text{number of steps}}$. Two geometric terms $k$ steps apart: their quotient is $r^k$, so take the $k$th root (if $k$ is even, $r$ can be either sign).
4. A negative ratio makes the signs alternate; a ratio between 0 and 1 makes the terms shrink.
5. Arithmetic series: $S_n = \frac{n(a_1 + a_n)}{2}$. Count the terms first: from $a$ to $b$ inclusive there are $b - a + 1$ integers.
6. Geometric series: use the formula, which the stem usually gives, $S_n = \frac{a_1(1 - r^n)}{1 - r}$, with $n$ equal to the number of terms. A term is not a sum.
7. Recursive rules: write the terms in order and label each one.
8. Cycles: divide the position by the cycle length. Remainder $r$ is the $r$th item; remainder 0 is the last item.