Sequences and Counting Reference

Sequences and Series

A sequence lists terms in order. A series adds those terms. Most precalculus problems ask us to recognize whether the pattern grows by repeated addition or repeated multiplication.

Solved Examples

Jump directly to a worked example by sequence or series type.

Fact Table

Pattern Signal Use
Arithmetic sequence Constant difference $d$. Linear explicit formula.
Geometric sequence Constant ratio $r$. Exponential explicit formula.
Finite arithmetic series Add arithmetic terms. Average first and last term, then multiply by count.
Finite geometric series Add geometric terms. Use the finite geometric sum formula.
Infinite geometric series Geometric series with $|r|<1$. The sum converges to a finite value.
Sigma notation $\sum$ with an index. Substitute each integer index value and add.

Content Formulas

Arithmetic Sequence
$$a_n=a_1+(n-1)d$$
Geometric Sequence
$$a_n=a_1r^{n-1}$$
Arithmetic Series
$$S_n=\frac n2(a_1+a_n)$$
Finite Geometric Series
$$S_n=a_1\frac{1-r^n}{1-r},\quad r\ne1$$
Infinite Geometric Series
$$S_\infty=\frac{a_1}{1-r},\quad |r|<1$$
Recursive Form
$$a_n=a_{n-1}+d$$ $$a_n=ra_{n-1}$$
Check the pattern before choosing a formula. Arithmetic means add the same amount; geometric means multiply by the same amount.

Classic Examples

Arithmetic Term

Find the 20th term of $7, 11, 15, 19,\ldots$.

Solution Steps
  1. Identify the first term and common difference.
  2. Use the arithmetic explicit formula for the twentieth term.
  3. State $a_{20}=83$.
$$a_1=7,\quad d=4$$ $$a_{20}=7+(20-1)4$$ $$a_{20}=83$$

Geometric Term

Find the 8th term of $3, 6, 12, 24,\ldots$.

Solution Steps
  1. Identify the first term and common ratio.
  2. Use the geometric explicit formula for the eighth term.
  3. State $a_8=384$.
$$a_1=3,\quad r=2$$ $$a_8=3(2)^{8-1}$$ $$a_8=384$$

Arithmetic Series

Find the sum of the first 30 terms of $5, 9, 13, 17,\ldots$.

Solution Steps
  1. Find the last term using the arithmetic sequence formula.
  2. Use the arithmetic series formula with the first and last terms.
  3. State the sum: $S_{30}=1890$.
$$a_{30}=5+(30-1)4=121$$ $$S_{30}=\frac{30}{2}(5+121)$$ $$S_{30}=1890$$

Infinite Geometric Series

Find the sum $12+6+3+\frac32+\cdots$.

Solution Steps
  1. Identify the first term and common ratio.
  2. Use the infinite geometric series formula.
  3. State the convergent sum: $S_\infty=24$.
$$a_1=12,\quad r=\frac12$$ $$S_\infty=\frac{12}{1-\frac12}$$ $$S_\infty=24$$