Geometric Sequences

A geometric sequence changes by a constant ratio and forms a discrete exponential pattern.

A sequence tracks terms; a series tracks their accumulating partial sums. Indexed terms appear along a horizontal progression above partial-sum markers. Shrinking term arrows and a target band distinguish terms approaching zero from partial sums actually converging.

In a geometric sequence, each term is produced by multiplying the previous term by the same factor.

Constant common ratio

A sequence is geometric when the ratio of consecutive nonzero terms is constant:

r=\frac{t_{n+1}}{t_n}.

For $3,6,12,24,\ldots$, each term is multiplied by $2$, so $r=2$. For $80,40,20,10,\ldots$, $r=1/2$.

If zeros occur, inspect the generating rule carefully because division by zero cannot calculate a ratio.

Recursive form

A geometric sequence can be defined recursively by

t_1=a,qquad t_n=rt_{n-1}\quad(n\ge2),

where $a$ is the first term and $r$ is the common ratio.

For $5,-10,20,-40,\ldots$,

t_1=5,qquad t_n=-2t_{n-1}.

The negative ratio makes signs alternate.

Worked example: find a distant term

Finding a common ratio from two terms

Suppose $t_3=18$ and $t_6=486$. Since three ratio steps separate the terms,

$$ 486=18r^3. $$

Thus $r^3=27$ and $r=3$. Then $18=ar^2=9a$, so $a=2$. The explicit rule is

t_n=2(3)^{n-1}.

Counting the number of steps between positions prevents an off-by-one exponent.

Solving for a term number

To find when $5(2)^{n-1}=640$, divide by $5$:

2^{n-1}=128=2^7.

Therefore $n-1=7$ and $n=8$. If the bases do not match conveniently, logarithms or a graph can solve for the exponent, followed by a domain check.

Common mistakes

Subtracting terms to test the pattern. Geometric sequences use ratios.

Using exponent $n$. Starting from $a$ requires $n-1$ ratio steps.

Confusing a percent change with a factor. A $12\%$ increase uses $1.12$; retaining $12\%$ uses $0.12$.

Ignoring a negative ratio. It causes alternating signs.

Accepting a fractional position. Sequence indices must follow the stated discrete domain.