Progressions
Arithmetic and geometric progressions: find a term, the difference or the ratio, the sum of the first terms and the sum of an infinite decreasing progression.
How to solve it
In an arithmetic progression each term is larger than the previous one by the same number $d$; in a geometric one, the same number of times $q$. Almost every problem comes down to counting the steps between two terms.
Step by step
- Find the kind: the same number added means arithmetic, the same number multiplied means geometric.
- Between $a_m$ and $a_n$ there are exactly $n - m$ steps: $a_n = a_m + (n - m)d$, $b_n = b_m \cdot q^{n - m}$.
- The sum of an arithmetic progression is the average of the first and the last term times the number of terms.
- The sum of a geometric one is $S_n = b_1 \frac{q^n - 1}{q - 1}$; an infinite sum exists when $|q| < 1$ and equals $\frac{b_1}{1 - q}$.
- If the number of terms is unknown, set up an equation and solve it.
Common mistakes
- Counting as many steps as terms: from $a_1$ to $a_7$ there are six steps, hence the $(n - 1)$.
- Counting the numbers from $57$ to $100$ as $100 - 57 = 43$; there are $44$, both ends included.
- Losing the sign of the ratio: with $q = -2$ the terms alternate in sign, and $q^n$ keeps its sign.
- Summing an infinite progression with $|q| \ge 1$: it has no finite sum.