Geometric Sequences

Learn about sequences where each term is multiplied by a constant ratio to get the next term.

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Definition

A geometric sequence is a sequence where each term is found by multiplying the previous term by a fixed number called the common ratio (rr).
If the first term is a1a_1 and the common ratio is rr, the sequence looks like:
a1,a1⋅r,a1⋅r2,a1⋅r3,…a_1, \quad a_1 \cdot r, \quad a_1 \cdot r^2, \quad a_1 \cdot r^3, \quad \ldots
The nth term formula is:
an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
To find the common ratio, divide any term by its previous term:
r=anan−1r = \frac{a_{n}}{a_{n-1}}

Try it now

Which of these is a geometric sequence?

Worked Examples

Is the sequence 3,6,12,24,48,…3, 6, 12, 24, 48, \ldots geometric? If so, find the common ratio.

1

Check the ratio between consecutive terms

63=2\frac{6}{3} = 2, 126=2\frac{12}{6} = 2, 2412=2\frac{24}{12} = 2, 4824=2\frac{48}{24} = 2 → All ratios are equal

2

Verify constant ratio

Since every ratio equals 22, this is geometric → Yes, it's geometric

3

State the common ratio

r=2r = 2 → The common ratio is 22

Common Mistakes

Confusing the exponent: using rnr^n instead of rn−1r^{n-1}

Why it's wrong: The first term has exponent 0: a1=a1⋅r0a_1 = a_1 \cdot r^0. So the nth term has exponent n−1n-1.

Correct: Always use an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}. For the 5th term, the exponent is 5−1=45-1=4.

Thinking a negative ratio means no geometric sequence

Why it's wrong: Geometric sequences can have negative ratios! The signs will alternate.

Correct: Example: 2,−6,18,−54,…2, -6, 18, -54, \ldots is geometric with r=−3r = -3.

Adding instead of multiplying to find the next term

Why it's wrong: This is the difference between arithmetic (add) and geometric (multiply) sequences.

Correct: In geometric sequences, always MULTIPLY by rr: if a3=12a_3 = 12 and r=2r = 2, then a4=12×2=24a_4 = 12 \times 2 = 24, NOT 12+2=1412 + 2 = 14.

Forgetting that rr can be a fraction (causing decay)

Why it's wrong: When 0<r<10 < r < 1, each term gets smaller. This models decay, not growth.

Correct: Example: 100,50,25,12.5,…100, 50, 25, 12.5, \ldots has r=0.5r = 0.5 (each term is half the previous).

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Practice Problems

16 problems
Problem 1 of 16
Easy

Which of these is a geometric sequence?

Why It Matters

Geometric sequences model many real-world phenomena:
  • Compound Interest: Your savings grow geometrically when interest is compounded
  • Population Growth: Bacteria double every hour, creating a geometric pattern
  • Depreciation: A car loses a percentage of its value each year
  • Physics: Bouncing balls lose a fraction of height with each bounce
  • Technology: Moore's Law describes geometric growth in computing power
Understanding geometric sequences is essential for financial planning, biology, physics, and computer science!

Real World Applications

Compound Interest

When money earns compound interest, the balance forms a geometric sequence where the common ratio is $(1 + \text{interest rate})$.

Example:

You invest 1000 euros at 5% annual interest. After each year, your balance is multiplied by 1.051.05: - Year 0: 1000 euros - Year 1: 1050 euros - Year 2: 1102.50 euros - Year 3: 1157.63 euros

1Try It Yourself

You deposit 500 euros in an account earning 8% annual interest.

How much will you have after 10 years?

Step 1: Write the mathematical expression

Use an=a1⋅rn−1a_n = a_1 \cdot r^{n-1} with a1=500a_1 = 500, r=1.08r = 1.08, n=11n = 11:

Bacterial Growth

Bacteria often double at regular intervals, creating a geometric sequence with $r = 2$.

Example:

If a culture starts with 100 bacteria and doubles every hour: - Hour 0: 100 bacteria - Hour 1: 200 bacteria - Hour 2: 400 bacteria - Hour 5: 100×25=3200100 \times 2^5 = 3200 bacteria

2Try It Yourself

A bacterial colony starts with 50 cells and triples every 2 hours.

How many bacteria will there be after 8 hours?

Step 1: Write the mathematical expression

Find the term number for 8 hours, then use the formula:

Depreciation

Cars and equipment lose value over time. If something loses a fixed percentage each year, it follows a geometric sequence with $r < 1$.

Example:

A car worth 20000 euros depreciates by 15% each year (keeps 85%): - Year 0: 20000 euros - Year 1: 17000 euros - Year 2: 14450 euros - Year 3: 12282.50 euros

3Try It Yourself

A computer worth 1200 euros loses 20% of its value each year.

What is it worth after 4 years?

Step 1: Write the mathematical expression

Use r=0.80r = 0.80 (keeping 80% each year):

Key Takeaways

  • 1A geometric sequence multiplies each term by a constant ratio rr to get the next term
  • 2The nth term formula is an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
  • 3Find the common ratio by dividing any term by its previous term: r=anan−1r = \frac{a_n}{a_{n-1}}
  • 4If ∣r∣>1|r| > 1, the sequence grows; if ∣r∣<1|r| < 1, it decays
  • 5If r<0r < 0, the terms alternate between positive and negative
  • 6Geometric sequences model compound interest, population growth, depreciation, and more

Frequently Asked Questions

In an arithmetic sequence, you ADD the same number to get the next term (common difference dd). In a geometric sequence, you MULTIPLY by the same number (common ratio rr). Example: 2,5,8,112, 5, 8, 11 is arithmetic (d=3d = 3); 2,6,18,542, 6, 18, 54 is geometric (r=3r = 3).
In an arithmetic sequence, you ADD the same number to get the next term (common difference dd). In a geometric sequence, you MULTIPLY by the same number (common ratio rr). Example: 2,5,8,112, 5, 8, 11 is arithmetic (d=3d = 3); 2,6,18,542, 6, 18, 54 is geometric (r=3r = 3).
Yes! When r<0r < 0, the terms alternate between positive and negative. For example, 2,−6,18,−54,…2, -6, 18, -54, \ldots has r=−3r = -3.
If r=1r = 1, every term is the same: a,a,a,a,…a, a, a, a, \ldots (constant sequence). If r=0r = 0, every term after the first is zero: a,0,0,0,…a, 0, 0, 0, \ldots
Divide consecutive terms. If all ratios are equal, it's geometric. For example, in 4,12,364, 12, 36: 124=3\frac{12}{4} = 3 and 3612=3\frac{36}{12} = 3, so it's geometric with r=3r = 3.

Glossary

Geometric sequence
A sequence where each term is obtained by multiplying the previous term by a constant ratio
Common ratio (rr)
The constant multiplier between consecutive terms
Exponential growth
When ∣r∣>1|r| > 1, causing the sequence to grow rapidly
Exponential decay
When ∣r∣<1|r| < 1, causing the sequence to shrink toward zero
Geometric progression
Another name for a geometric sequence

Formula Card

nth Term

an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}

Find any term given the first term and common ratio

Common Ratio

r=anan−1r = \frac{a_n}{a_{n-1}}

Find the ratio by dividing consecutive terms

Recursive Formula

an=an−1⋅ra_n = a_{n-1} \cdot r

Each term equals the previous term times $r$

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