Identifying a Geometric Sequence

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

Check the ratio between consecutive terms: $\frac{6}{3} = 2$, $\frac{12}{6} = 2$, $\frac{24}{12} = 2$, $\frac{48}{24} = 2$ = All ratios are equal

Verify constant ratio: Since every ratio equals $2$, this is geometric = Yes, it's geometric

State the common ratio: $r = 2$ = The common ratio is $2$

Answer: Yes, this is a geometric sequence with common ratio $r = 2$. Each term is double the previous term.

Finding the nth Term

Find the 8th term of the geometric sequence: $5, 15, 45, 135, \ldots$

Identify the first term: $a_1 = 5$ = First term is $5$

Find the common ratio: $r = \frac{15}{5} = 3$ = Common ratio is $3$

Apply the nth term formula: $a_n = a_1 \cdot r^{n-1}$ $a_8 = 5 \cdot 3^{8-1}$ $a_8 = 5 \cdot 3^7$ = Set up the formula

Calculate: $a_8 = 5 \cdot 2187 = 10935$ = $a_8 = 10935$

Answer: The 8th term is $10935$.

Geometric Sequence with Fractional Ratio

A ball is dropped from 256 cm. Each bounce reaches $\frac{1}{2}$ of the previous height. What height does it reach after the 5th bounce?

Identify the sequence parameters: Initial height: $a_1 = 256$ cm Common ratio: $r = \frac{1}{2}$ = Setup complete

Determine which term we need: After 5th bounce = 6th term (including initial drop) = Need $a_6$

Apply the formula: $a_6 = 256 \cdot \left(\frac{1}{2}\right)^{6-1}$ $a_6 = 256 \cdot \left(\frac{1}{2}\right)^5$ = Formula set up

Calculate: $a_6 = 256 \cdot \frac{1}{32} = 8$ cm = $a_6 = 8$ cm

Answer: After the 5th bounce, the ball reaches a height of $8$ cm.

Finding the Common Ratio Given Two Terms

In a geometric sequence, $a_3 = 24$ and $a_6 = 192$. Find the common ratio and the first term.

Write equations using the nth term formula: $a_3 = a_1 \cdot r^2 = 24$ $a_6 = a_1 \cdot r^5 = 192$ = Two equations

Divide the equations to eliminate $a_1$: $\frac{a_6}{a_3} = \frac{a_1 \cdot r^5}{a_1 \cdot r^2} = \frac{192}{24}$ $r^3 = 8$ = $r^3 = 8$

Solve for r: $r = \sqrt[3]{8} = 2$ = $r = 2$

Find $a_1$ using $a_3 = 24$: $a_1 \cdot 2^2 = 24$ $a_1 \cdot 4 = 24$ $a_1 = 6$ = $a_1 = 6$

Answer: The common ratio is $r = 2$ and the first term is $a_1 = 6$. The sequence is $6, 12, 24, 48, 96, 192, \ldots$

Mistake: Confusing the exponent: using $r^n$ instead of $r^{n-1}$

Why: The first term has exponent 0: $a_1 = a_1 \cdot r^0$. So the nth term has exponent $n-1$.

Correct: Always use $a_n = a_1 \cdot r^{n-1}$. For the 5th term, the exponent is $5-1=4$.

Mistake: Thinking a negative ratio means no geometric sequence

Why: Geometric sequences can have negative ratios! The signs will alternate.

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

Mistake: Adding instead of multiplying to find the next term

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

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

Mistake: Forgetting that $r$ can be a fraction (causing decay)

Why: When $0 < r < 1$, each term gets smaller. This models decay, not growth.

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

Compound Interest

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

You invest 1000 euros at 5% annual interest. After each year, your balance is multiplied by $1.05$: - Year 0: 1000 euros - Year 1: 1050 euros - Year 2: 1102.50 euros - Year 3: 1157.63 euros

Bacterial Growth

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

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: $100 \times 2^5 = 3200$ bacteria

Depreciation

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

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

Q: What is the difference between arithmetic and geometric sequences?

A: In an arithmetic sequence, you ADD the same number to get the next term (common difference $d$). In a geometric sequence, you MULTIPLY by the same number (common ratio $r$). Example: $2, 5, 8, 11$ is arithmetic ($d = 3$); $2, 6, 18, 54$ is geometric ($r = 3$).

Q: Can the common ratio be negative?

A: Yes! When $r < 0$, the terms alternate between positive and negative. For example, $2, -6, 18, -54, \ldots$ has $r = -3$.

Q: What happens when $r = 1$ or $r = 0$?

A: If $r = 1$, every term is the same: $a, a, a, a, \ldots$ (constant sequence). If $r = 0$, every term after the first is zero: $a, 0, 0, 0, \ldots$

Q: How do I know if a sequence is geometric?

A: Divide consecutive terms. If all ratios are equal, it's geometric. For example, in $4, 12, 36$: $\frac{12}{4} = 3$ and $\frac{36}{12} = 3$, so it's geometric with $r = 3$.