Revision sheet

Sequences

• Explicit: $a_n$ straight from $n$.
• Recursive: a first term and a rule from $a_n$ to $a_{n+1}$.

Watch out: check your formula with $n = 1$.

Arithmetic sequences

• Add $d$ each time: $a_n = a_1 + (n - 1)d$.
• $d = \dfrac{a_q - a_p}{q - p}$.

Watch out: $n - 1$ steps from $a_1$ to $a_n$.

Geometric sequences

• Multiply by $r$ each time: $a_n = a_1 r^{\,n-1}$.
• $r$ = a term divided by the one before.

Watch out: the exponent is $n - 1$.

Arithmetic series

• $\displaystyle\sum_{k=p}^{q}$ has $q - p + 1$ terms.
• $S_n = \dfrac{n(a_1 + a_n)}{2}$.
• $1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}$.

Watch out: count the terms with $+1$.

Finite geometric series

• $S_n = a_1\,\dfrac{r^n - 1}{r - 1}$.
• $n$ is the number of terms; the power is $r^n$.

Watch out: the last term is $a_1 r^{\,n-1}$, but the sum uses $r^n$.

Infinite geometric series

• $|r| < 1$: $S = \dfrac{a}{1 - r}$.
• $|r| \ge 1$: no sum.
• Repeating decimals are such series.

Watch out: check $|r| < 1$ before using the formula.

The binomial theorem

• $(a + b)^n = \sum \dbinom{n}{k} a^{\,n-k} b^{\,k}$.
• Coefficients: a row of Pascal's triangle.
• Include the powers of the constants.

Watch out: $(a + b)^2 \ne a^2 + b^2$.