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Growth & Decay

Finite Geometric Series

Sum sequences that grow (or shrink) by multiplication using powerful algebraic techniques

Key Takeaways
Two formula forms: $S_n = \frac{a(r^n - 1)}{r - 1}$ when $r > 1$, or $S_n = \frac{a(1 - r^n)}{1 - r}$ when $r < 1$
Derivation technique: Multiply the series by $r$, subtract, and watch middle terms cancel
Growth vs decay: $r > 1$ creates explosive growth, $r < 1$ creates gradual decay

Multiplying to Add

A geometric series is the sum of terms in a geometric sequence, where each term is found by multiplying the previous term by a constant ratio.

For example, the sequence 2, 6, 18, 54 (multiplying by 3 each time) has the series:

$$2 + 6 + 18 + 54 = 80$$

But how do we find the sum of 50 or 100 terms? We need a formula that works for any number of terms. The brilliant technique is to multiply the entire series by the common ratio, then subtract to eliminate all but two terms!

Notation
We denote the sum of the first $n$ terms as $S_n$. The general term is $T_n = ar^{n-1}$ where $a$ is the first term and $r$ is the common ratio.

The Sum Formulas

The sum of a finite geometric series can be calculated using one of two equivalent formulas, depending on whether the ratio is greater than or less than 1:

When r > 1 (Growth)
$$S_n = \frac{a(r^n - 1)}{r - 1}$$
where $a$ = first term
$r$ = common ratio ($r > 1$)
$n$ = number of terms
When r < 1 (Decay)
$$S_n = \frac{a(1 - r^n)}{1 - r}$$
where $a$ = first term
$r$ = common ratio ($r < 1$)
$n$ = number of terms
Important Restriction
Both formulas require $r \neq 1$. When $r = 1$, the sequence is constant (all terms equal $a$), so $S_n = na$.

Deriving the Formula

The geometric series formula comes from a clever algebraic trick. Follow each step to see how multiplying by $r$ and subtracting causes all the middle terms to disappear!

Deriving the Formula
The multiplication-and-subtraction method
Step 1: Write the series
$$S_n = a + ar + ar^2 + ar^3 + \dots + ar^{n-1}$$
Start with the general form of a geometric series.
Step 1 of 5

Interactive Formula Explorer

Experiment with different values to see how the formula works. Notice how both formula forms give the same result!

Geometric Sum Calculator
Explore how the sum changes with different parameters
Sequence:
2.00
6.00
18.00
54.00
162.00
Formula Used:
$$S_n = \frac{a(r^n - 1)}{r - 1}$$
$S_{5} = \frac{2(3^{5} - 1)}{3 - 1}$
Sum = 242.00
Verification (adding all terms):
242.00

Growth vs Decay

Geometric series behave very differently depending on whether the common ratio is greater than or less than 1. Watch the dramatic difference!

Growth vs Decay Series
Compare exponential growth (r > 1) with exponential decay (r < 1)
Growth (r = 1.5)
Term 1: 10.00
Term 2: 15.00
Term 3: 22.50
Term 4: 33.75
Term 5: 50.63
Term 6: 75.94
Term 7: 113.91
Term 8: 170.86
Sum:
492.58
Decay (r = 0.6)
Term 1: 10.00
Term 2: 6.00
Term 3: 3.60
Term 4: 2.16
Term 5: 1.30
Term 6: 0.78
Term 7: 0.47
Term 8: 0.28
Sum:
24.58
Key Observation
Growth series (r > 1) explode rapidly, while decay series (r < 1) converge toward a limit. This is why we use different formula forms for each case!

Practice Questions

Additional Practice Questions
Access a printable set of extended problems with full solutions and PDF export options for both blank and answered versions.
View Practice Page

Test your understanding with these practice problems, organized by difficulty level.

Question 1
Question 2
Question 3

Self-Check Quiz

Question 1
What is the sum of $2 + 4 + 8 + 16 + 32$?
Question 2
For a geometric series with $a = 5$, $r = 0.5$, and $n = 6$, which formula should you use?
Question 3
What happens to the geometric series formula when $r = 1$?
Question 4
In the derivation of the geometric series formula, what key step eliminates the middle terms?
Question 5
If $\\sum_{k=1}^{5} 3(2)^{k-1} = S$, what is $S$?
What's Next?

Now that you've mastered finite geometric series, you're ready to explore infinite geometric series and discover when they converge to a finite sum!