What is Geometric Sequence

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Geometric Sequences

Geometric sequences are those in which the following number is obtained by multiplying the previous term by a constant known as the common ratio. The letter $r$ is used to show the common ratio.

We have an increasing geometric sequence if $r$ is more than $1$. We have a decreasing geometric sequence if $r$ is between $0$ and $1$.

The Geometric Sequences Formula

We can use the geometric sequence formula to find any number in the sequence.

$a_n = a \ r^{(n-1)}$, where:

$a$: The first term
$a_n$: The $n$th term
$n$: The term poistion
$r$: Common ratio

Finding the terms in a Geometric Sequence

If we divide any term by the previous term, we can find the common ratio: $r \ = \ \frac{a_n}{a_{n \ - 1}}$

Example

Consider the geometric sequence: $2, \ 6, \ 18, \ 54, \ 162, \ ...$, Find $a_{7} \ = \ ?$

Solution

Before using the geometric sequence formula, we need to know the first term, the common ratio, and the position of the term we want to find:

The first term: $a \ = \ 2$
common ratio: $r \ = \ 3$
Term's position: $n \ = \ 7$

Now, we fill in the formula with these numbers:

$a_n = a \ r^{(n-1)} \ ⇒ \ a_7 = 2 \ 3^{(6)} \ = \ 2 \times 729 \ = \ 1458$

We can see that the number is very big. Depending on what the common ratio is, geometric sequences tend to grow quickly.

The formula for the Sum of a Geometric Sequence

With the sum of a finite geometric sequence formula, you can find the sum of a geometric sequence's first $n$ terms. Consider a geometric series with $n$ terms, where the first term is $a$, and the ratio $r$: $a, \ ar, \ ar^2, \ ar^3, \ ... \ , \ ar^{n \ - \ 1}$

The sum is shown by $S_n$ and given by the formula:

$S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1}$, when $r \ ≠ \ 1$
$S_n \ = \ na$, when $r \ = \ 1$

Example

Add up the first four terms in: $10, \ 30, \ 90, \ 270, \ 810, \ 2430, \ ...$

Solution

The first term: $a \ = \ 10$
common ratio: $r \ = \ 3$
$n \ = \ 4$

So:

$S_4 \ = \ \frac{10(3^4 \ - \ 1)}{3 \ - \ 1} \ = \ \frac{10(80)}{2} \ = \ 400$

Geometric Sequences

Think of this lesson as more than a rule to memorize. Geometric Sequences is about patterns, terms, common differences, ratios, and sums. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

For arithmetic sequences, use $a_n=a_1+(n-1)d$. For geometric sequences, use $a_n=a_1r^{n-1}$. The pattern tells you which formula fits.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Decide whether the pattern adds or multiplies.
Identify the first term and the difference or ratio.
Choose the term or sum formula.
Check whether the answer makes sense in the pattern.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Geometric Sequences

1) Find the explicit formula: $3, \ 6, \ 12, \ 24, \ ...$

2) Find the explicit formula: $81, \ 27, \ 9, \ 3, \ ...$

3) Find the explicit formula: $-2, \ 10, \ -50, \ 250, \ ...$

4) Find $a_6$ if $a_1 \ = \ 5$ and $r \ = \ 3$

5) Find $a_8$ if $a_1 \ = \ 64$ and $r \ = \ \frac{1}{2}$

6) Find $a_6$ if $a_3 \ = \ 18$ and $r \ = \ 3$

7) Find $a_9$: $7, \ -14, \ 28, \ -56, \ ...$

8) Find $a_{10}$: $160, \ 80, \ 40, \ 20, \ ...$

9) If $a_2 \ = \ 12$ and $a_5 \ = \ 324$, find $a_7$, assuming $r$ is positive

10) Insert three geometric means between $2$ and $162$, using a positive ratio

11) Find the sum of the first $6$ terms of $4, \ 12, \ 36, \ ...$

12) Find the sum of the first $5$ terms of $96, \ 48, \ 24, \ ...$

13) Find $S_8$ if $a_1 \ = \ -3$ and $r \ = \ 2$

14) Find $n$ if $3645$ is a term of $5, \ 15, \ 45, \ ...$

15) Find $a_7$ if $a_4 \ = \ 54$ and $r \ = \ -3$

16) A car worth $\$20000$ keeps $80\%$ of its value each year. What is its value at the start of year $5$?

17) If $a_2 \ = \ 6$ and $a_6 \ = \ 96$, find $a_{10}$, assuming $r$ is positive

18) Find the sum of the first $10$ terms of $2, \ -4, \ 8, \ -16, \ ...$

19) Find $a_8$ if $a_1 \ = \ \frac{1}{3}$ and $r \ = \ \frac{3}{2}$

20) Find $n$ if $3 \ + \ 6 \ + \ 12 \ + \ \cdots \ + \ a_n \ = \ 381$

Answers

1)Find the explicit formula: $3, \ 6, \ 12, \ 24, \ ...$