How to Solve Infinite Geometric Series

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Infinite Geometric Series

A geometric series with an infinite number of terms is called an infinite geometric series. The infinite geometric series is shown as \(a, \ ar, \ ar^2, \ ar^3, \ ... \ ,\) to \(∞\).

Infinite Geometric Formula

If the value of \(r\) is between \(-1\) and \(1\), then the formula for an infinite series is: \(S_n \ = \ \frac{a}{1 \ - \ r}\)
Where,

\(a \ =\) The first term
\(r \ =\) The common ratio

Only if \(|r| \ < \ 1\) does the geometric series add up to a finite sum. If \(|r| \ ≥ \ 1\), the series doesn't converge, and there is no finite sum.

Example

The geometric series is \(2, \ 8, \ 32, \ 128, \ ...\)

To find the sum of \(2, \ 8, \ 32, \ 128, \ ...,\) first we need to find the value of \(r\): \(r \ = \ \frac{8}{2} \ = \ 4\)

Since \(r \ > \ 1\), the series does not converge and has no sum.

Infinite Geometric Series

Think of this lesson as more than a rule to memorize. Infinite Geometric Series 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 Infinite Geometric Series

1) Determine whether the series converges or diverges: \(4, \ 12, \ 36, \ 108, \ ...\)

2) Determine whether the series converges or diverges, and find the sum if it converges: \(10, \ 5, \ \frac{5}{2}, \ ...\)

3) Determine whether the series converges or diverges, and find the sum if it converges: \(9, \ -3, \ 1, \ -\frac{1}{3}, \ ...\)

4) Determine whether the series converges or diverges: \(7, \ 14, \ 28, \ 56, \ ...\)

5) Determine whether the series converges or diverges, and find the sum if it converges: \(100, \ 80, \ 64, \ ...\)

6) Determine whether the series converges or diverges: \(6, \ -9, \ 13.5, \ -20.25, \ ...\)

7) Find the sum, if it exists: \(3 \ + \ 1 \ + \ \frac{1}{3} \ + \ \cdots\)

8) Find the sum, if it exists: \(5 \ - \ 2.5 \ + \ 1.25 \ - \ \cdots\)

9) Find the sum, if it exists: \(\frac{1}{4} \ + \ \frac{1}{8} \ + \ \frac{1}{16} \ + \ \cdots\)

10) Find the sum, if it exists: \(81 \ + \ 27 \ + \ 9 \ + \ \cdots\)

11) Find the sum, if it exists: \(-12 \ - \ 6 \ - \ 3 \ - \ \cdots\)

12) Determine whether the series converges or diverges: \(2 \ + \ 2 \ + \ 2 \ + \ \cdots\)

13) Find the sum, if it exists: \(18 \ - \ 6 \ + \ 2 \ - \ \frac{2}{3} \ + \ \cdots\)

14) Find the sum, if it exists: \(64 \ + \ 16 \ + \ 4 \ + \ \cdots\)

15) Find the infinite sum if \(a_1 \ = \ 15\) and \(r \ = \ -\frac{2}{5}\)

16) Determine whether the series with \(a_1 \ = \ 3\) and \(r \ = \ 1.2\) converges or diverges

17) Write \(0.777\ldots\) as a fraction using an infinite geometric series

18) A ball rebounds \(12\) feet, then each rebound is \(\frac{3}{4}\) of the previous rebound. What is the total rebound distance?

19) An infinite geometric series has first term \(10\) and sum \(40\). Find \(r\)

20) Evaluate: \(\sum_{n \ = \ 1}^{\infty} 6\left(-\frac{2}{3}\right)^{n \ - \ 1}\)

Answers

1)Determine whether the series converges or diverges: \(4, \ 12, \ 36, \ 108, \ ...\)