5.1: Arithmetic and Geometric Sequences | Algebra 2 Honors

How it works

An arithmetic sequence adds a fixed difference from one term to the next. A geometric sequence multiplies by a fixed ratio. For a nonzero geometric starting term, ratios between successive terms reveal the pattern; differences reveal arithmetic patterns. Sequence inputs are discrete integer positions.

a_n=a_1+(n-1)d,\qquad a_n=a_1r^{n-1}

Worked examples

Example 1

Find term $\displaystyle 6$ of $\displaystyle 4,7,10,\ldots$ .

The difference is $\displaystyle 3$ . Thus $\displaystyle a_6=4+5(3)=19$ .

Example 2

Find term $\displaystyle 5$ of $\displaystyle 3,6,12,\ldots$ .

The ratio is $\displaystyle 2$ . Thus $\displaystyle a_5=3\cdot2^4=48$ .

Problems

Problem 5.1.1

Find term $\displaystyle 8$ of $\displaystyle 10,7,4,\ldots$ .

Show worked solution

The difference is $\displaystyle -3$ , so $\displaystyle a_8=10+7(-3)=-11$ .

Problem 5.1.2

Find term $\displaystyle 4$ of $\displaystyle 81,27,9,\ldots$ .

Show worked solution

The ratio is $\displaystyle 1/3$ . Thus $\displaystyle a_4=81(1/3)^3=3$ .

Problem 5.1.3

The graph shows terms of $\displaystyle a_n=2\cdot3^{n-1}$ . Give its first four terms and explain why the graph uses separate points.

Four separate points represent 2, 6, 18, and 54 at sequence positions 1 through 4. — A sequence has integer positions rather than a continuous input interval.

Show worked solution

The terms are $\displaystyle 2,6,18,54$ . Only integer positions $\displaystyle n=1,2,3,\ldots$ belong to the sequence, so the graph is discrete.

Four separate points represent 2, 6, 18, and 54 at sequence positions 1 through 4. The common ratio is 3; the first four terms are 2, 6, 18, 54. — The common ratio is 3; the first four terms are 2, 6, 18, 54.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 5.1.1

Problem 5.1.2

Problem 5.1.3

Need help with this topic?