How to Solve Finite Geometric Series

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Finite Geometric Series

There are a finite set of numbers in a finite geometric series. This means there will be a first and last term for the series. Finite geometric series are convergent.

Finite Geometric Formula

Use the formula to find the sum of a finite geometric series. $S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1}$, when $r \ ≠ \ 1$

Where $a$ is the first term, $n$ is the number of terms, and $r$ is the common ratio.

Example

Find the total of the first $6$ terms of the geometric series if $a \ = \ 5$ and $r \ = \ 3$.

$S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ ⇒ \ \frac{5(3^6 \ - \ 1)}{3 \ - \ 1} \ = \ 1820$

Finite Geometric Series

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

1) Evaluate: $2 \ + \ 4 \ + \ 8 \ + \ 16 \ + \ 32$

2) Evaluate: $5 \ + \ 15 \ + \ 45 \ + \ 135$

3) Find the sum of the first $6$ terms if $a_1 \ = \ 3$ and $r \ = \ 2$

4) Find the sum of the first $5$ terms if $a_1 \ = \ 7$ and $r \ = \ 3$

5) Evaluate: $\sum_{k \ = \ 1}^{4} 6(2)^{k \ - \ 1}$

6) Evaluate: $\sum_{k \ = \ 1}^{5} 10\left(\frac{1}{2}\right)^{k \ - \ 1}$

7) Evaluate: $81 \ + \ 27 \ + \ 9 \ + \ 3 \ + \ 1$

8) Find $S_6$ for $4, \ -8, \ 16, \ -32, \ ...$

9) Evaluate: $\sum_{k \ = \ 1}^{7} 5(-3)^{k \ - \ 1}$

10) Find the sum of the first $8$ terms of $96, \ 48, \ 24, \ ...$

11) Find the sum of the first $10$ terms of $1, \ 4, \ 16, \ ...$

12) Evaluate: $\sum_{k \ = \ 0}^{5} 12\left(\frac{1}{3}\right)^k$

13) Find the sum of the first $9$ terms of $2, \ -6, \ 18, \ ...$

14) Find the sum of $7$ terms if $a_3 \ = \ 20$ and $r \ = \ 2$

15) Evaluate $3 \ + \ 6 \ + \ 12 \ + \ \cdots \ + \ 384$

16) A savings plan deposits $\$25$, then doubles each deposit for $10$ deposits. What is the total deposited?

17) Evaluate: $\sum_{k \ = \ 1}^{6} 64\left(-\frac{1}{2}\right)^{k \ - \ 1}$

18) Find the sum of the first $12$ terms of $500, \ 250, \ 125, \ ...$

19) A finite geometric series has $a_1 \ = \ 9$, $r \ = \ 4$, and $S_n \ = \ 3069$. Find $n$

20) Evaluate: $\sum_{k \ = \ 2}^{8} 3(2)^k$

Answers

1)Evaluate: $2 \ + \ 4 \ + \ 8 \ + \ 16 \ + \ 32$