Determine terms of a geometric sequence

A geometric sequence is understood as an ordered list in which each term is obtained by multiplying the preceding term by a constant common ratio, allowing positive, negative, fractional, and zero ratios where appropriate. Learners determine terms from a given first term, common ratio, recursive rule, or explicit rule $a_n=a_1r^{n-1}$ , interpret the role of the index, and distinguish multiplicative change from the additive change of an arithmetic sequence; this scope excludes complex ratios and advanced infinite-series analysis.

Detailed Explanation: Determine terms of a geometric sequence

A geometric sequence is a list of numbers in which each term is found by multiplying the previous term by the same number. This constant multiplier is called the common ratio, written as $r$ .

If the first term is $a_1$ and the common ratio is $r$ , use

a_n=a_1r^{n-1}

The index $n$ tells you which term to find. The exponent is $n-1$ because the first term has been multiplied by the ratio zero times.

Example

Find the first five terms and the fifth term of the geometric sequence with first term $a_1=3$ and common ratio $r=-2$ .

Step 1: Generate the terms by multiplying by $-2$ .

Start with $3$ :

\begin{aligned} a_1&=3\\ a_2&=3(-2)=-6\\ a_3&=(-6)(-2)=12\\ a_4&=12(-2)=-24\\ a_5&=(-24)(-2)=48 \end{aligned}

So the first five terms are

3,\ -6,\ 12,\ -24,\ 48.

Step 2: Use the explicit formula to check the fifth term.

Since $a_1=3$ , $r=-2$ , and $n=5$ ,

a_5=a_1r^{5-1}

a_5=3(-2)^4

a_5=3(16)=48.

Therefore, the fifth term is

\boxed{48}.

The sequence changes multiplicatively because each term is multiplied by $-2$ . This is different from an arithmetic sequence, where the same number is added or subtracted each time.

Learn by doing: Determine terms of a geometric sequence

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Patterning - First Values from Equation for Geometric Pattern

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