9.1: Sequences and Series | Precalculus

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 9.1.1

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 9.1.2

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 9.1.3

Find the sum of the first five terms of the geometric sequence $\displaystyle 3,6,12,\ldots$ .

Show worked solution

The terms are $\displaystyle 3,6,12,24,48$ . Their sum is $\displaystyle 93$ , also given by $\displaystyle 3(1-2^5)/(1-2)$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 9.1.1

Problem 9.1.2

Problem 9.1.3

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