Geometric Sequences

🔁 Geometric Sequences

Part 1 of 5 — The Common Ratio

Topics in This Part

🔑 Key Concept: In a geometric sequence you get from each term to the next by multiplying by the same number — the common ratio $r$. (Arithmetic sequences add; geometric sequences multiply.)

What Makes a Sequence Geometric?

A sequence is geometric when each term is the previous term times a fixed number $r$:

$2,\; 6,\; 18,\; 54,\; 162,\; \dots \qquad (\times 3 \text{ each time})$

Here $r = 3$. To find $r$, divide any term by the term before it:

$r = \frac{a_2}{a_1} = \frac{6}{2} = 3, \qquad r = \frac{a_3}{a_2} = \frac{18}{6} = 3 \;\checkmark$

The ratio must be the same between every consecutive pair — that's the test.

The Ratio Can Be Anything Nonzero

💡 A negative ratio makes the signs flip $+,-,+,-$. A ratio between $-1$ and $1$ makes terms shrink toward $0$.

Concept Check 🎯

A Reliable Way to Get $r$

You never have to guess the ratio — just divide consecutive terms:

$r = \frac{\text{any term}}{\text{the term before it}}$

If you get the same answer for every pair, the sequence is geometric and that number is $r$. If the answers differ, it isn't geometric at all.

Find the Common Ratio 🧮

Divide any term by the previous one. (Fractions like 1/3 are fine.)

1) $7,\,14,\,28,\,56,\dots \qquad r = \,?$ 2) $81,\,27,\,9,\,3,\dots \qquad r = \,?$ 3) $2,\,-10,\,50,\,-250,\dots \qquad r = \,?$

Geometric vs. Arithmetic — The One-Question Test

Ask: "To get the next term, do I add the same amount or multiply by the same amount?"

• Add the same amount → arithmetic (common difference $d$).
• Multiply by the same amount → geometric (common ratio $r$).

Some sequences (like the perfect squares $1,4,9,16$) do neither — the gaps and the ratios both keep changing.

Geometric or Arithmetic? 🔽

Decide what kind of sequence each one is.

Recap

• A geometric sequence has a constant common ratio $r$.
• Find it by dividing any term by the previous term: $r = \dfrac{a_{n}}{a_{n-1}}$.
• $r > 1$ grows, $0 < r < 1$ shrinks, $r < 0$ alternates sign.

In Part 2 we turn $r$ into a formula that jumps straight to any term — no listing required.

🔁 Geometric Sequences

Part 2 of 5 — The Explicit Formula

🔑 The Idea: To reach the $n$th term you start at $a_1$ and multiply by $r$ a total of $(n-1)$ times. That's exactly what the explicit formula says.

The Explicit (nth-Term) Formula

$\boxed{\,a_n = a_1 \cdot r^{\,n-1}\,}$

• $a_1$ — the first term
• $r$ — the common ratio
• $n$ — the term number you want

The exponent is $n-1$ (not $n$) because the first term has been multiplied by $r$ zero times: $a_1 = a_1 \cdot r^{0}$.

Worked Example: $3,\,6,\,12,\,24,\dots$

Here $a_1 = 3$ and $r = 2$, so the rule is:

$a_n = 3 \cdot 2^{\,n-1}$

Find the 7th term:

$a_7 = 3 \cdot 2^{\,7-1} = 3 \cdot 2^{6} = 3 \cdot 64 = 192$

✅ Check by listing: $3,6,12,24,48,96,192$ — the 7th term is indeed $192$. ✓

Worked Example with a Fractional Ratio

Sequence $80,\,40,\,20,\,10,\dots$ has $a_1 = 80$ and $r = \dfrac{1}{2}$:

$a_n = 80 \cdot \left(\tfrac{1}{2}\right)^{\,n-1}$

The 5th term:

$a_5 = 80 \cdot \left(\tfrac{1}{2}\right)^{4} = 80 \cdot \tfrac{1}{16} = 5$

⚠️ Watch the exponent. For the 5th term the exponent is $5-1 = 4$, not $5$. Subtracting one is the single most common slip.

Concept Check 🎯

Building a Formula From a List

Given a sequence, writing its explicit formula is always the same two steps:

1. Read off $a_1$ — the very first term.
2. Divide to get $r$ — second term over first term.

Then drop both into $a_n = a_1 \cdot r^{\,n-1}$. Let's build one piece by piece.

Build the Formula 🔽

The sequence is $4,\,12,\,36,\,108,\dots$

Mind the Exponent

Before the next drill, lock in the rule that trips up most students:

⚠️ The exponent is $\mathbf{n-1}$, not $n$. The $1^{\text{st}}$ term uses $r^{0}=1$, the $2^{\text{nd}}$ uses $r^{1}$, the $5^{\text{th}}$ uses $r^{4}$, and so on — always one less than the term number.

Use the Formula 🧮

Apply $a_n = a_1 \cdot r^{\,n-1}$.

1) $a_1 = 2,\; r = 3$. Find $a_5$. 2) $a_1 = 6,\; r = 2$. Find $a_4$. 3) $a_1 = 100,\; r = \tfrac{1}{5}$. Find $a_3$.

🔁 Geometric Sequences

Part 3 of 5 — Recursive Rules, Missing Terms & Geometric Mean

🔑 Two ways to describe the same sequence: the explicit rule jumps to any term, while the recursive rule says how to get the next term from the one before it.

Recursive Form

A recursive rule needs a starting value and a step:

$a_1 = \text{(given)}, \qquad a_n = r \cdot a_{n-1}$

For $5,\,15,\,45,\,135,\dots$ ($r = 3$):

$a_1 = 5, \qquad a_n = 3\,a_{n-1}$

💡 Both describe the same sequence. Recursive needs the previous term; explicit only needs $n$.

Finding $r$ When Terms Aren't Adjacent

If you know two terms that are $k$ apart, then $\dfrac{a_{m+k}}{a_m} = r^{\,k}$.

Example: $a_1 = 4$ and $a_4 = 108$. Find $r$.

There are $4 - 1 = 3$ multiplications between them:

$a_4 = a_1 \cdot r^{3} \;\Rightarrow\; 108 = 4 r^{3} \;\Rightarrow\; r^{3} = 27 \;\Rightarrow\; r = \sqrt[3]{27} = 3$

So the sequence is $4,\,12,\,36,\,108,\dots$ ✓

The Geometric Mean (Missing Middle Term)

The geometric mean of two positive numbers $a$ and $b$ is the term that sits between them in a geometric sequence:

$\text{geometric mean} = \sqrt{a \cdot b}$

Example: Fill the blank $4,\,\underline{\;\;?\;\;},\,9$

$\sqrt{4 \cdot 9} = \sqrt{36} = 6$

So the sequence is $4,\,6,\,9$ with $r = \dfrac{6}{4} = \dfrac{3}{2}$ (and $\dfrac{9}{6} = \dfrac{3}{2}$ ✓).

⚠️ Use the geometric mean $\sqrt{ab}$ here — not the average $\frac{a+b}{2}$. The average of $4$ and $9$ is $6.5$, which would not keep the ratio constant.

Match the Form 🔽

Use the sequence $6,\,18,\,54,\,162,\dots$

Recursive Rules Always Multiply

A common mistake is to write a recursive geometric rule as $a_n = a_{n-1} + r$ — that's arithmetic. Geometric rules always multiply by $r$:

$a_n = r \cdot a_{n-1}$

Keep that straight and the next check is easy.

Concept Check 🎯

Before You Drill

The next problems blend two skills:

• Geometric mean → $\sqrt{a\cdot b}$ for the term between $a$ and $b$.
• Solving for $r$ → if $a_3 = a_1 r^{2}$, then $r = \sqrt{a_3 / a_1}$ (take the positive root unless told otherwise).

Match each part to the right tool, then compute.

Mixed Drill 🧮

1) Geometric mean of $5$ and $20$ $= \,?$ 2) $a_1 = 3,\; a_3 = 75$. Find $r$ (take the positive value). 3) Fill the blank: $2,\,\underline{\;\;?\;\;},\,18$ (positive term).

🔁 Geometric Sequences

Part 4 of 5 — Finite Geometric Series

🔑 A series is the sum of a sequence's terms. Adding $30$ terms by hand is brutal — the geometric-sum formula does it in one line.

The Finite Sum Formula

The sum of the first $n$ terms of a geometric sequence is:

$\boxed{\,S_n = a_1 \cdot \frac{1 - r^{\,n}}{1 - r}\,}, \qquad r \ne 1$

• $a_1$ — first term
• $r$ — common ratio
• $n$ — how many terms you're adding

Worked Example: Sum the first 6 terms of $3,\,6,\,12,\dots$

Here $a_1 = 3,\; r = 2,\; n = 6$:

$S_6 = 3 \cdot \frac{1 - 2^{6}}{1 - 2} = 3 \cdot \frac{1 - 64}{-1} = 3 \cdot \frac{-63}{-1} = 3 \cdot 63 = 189$

✅ Check by listing: $3+6+12+24+48+96 = 189$. ✓

Worked Example with $r < 1$

Sum the first $4$ terms of $16,\,8,\,4,\,2,\dots$, where $a_1 = 16,\; r = \dfrac{1}{2}$:

$S_4 = 16 \cdot \frac{1 - \left(\tfrac{1}{2}\right)^{4}}{1 - \tfrac{1}{2}} = 16 \cdot \frac{1 - \tfrac{1}{16}}{\tfrac{1}{2}} = 16 \cdot \frac{\tfrac{15}{16}}{\tfrac{1}{2}}$

$= 16 \cdot \frac{15}{16} \cdot 2 = 15 \cdot 2 = 30$

✅ Check: $16+8+4+2 = 30$. ✓

💡 Dividing by $\tfrac{1}{2}$ is the same as multiplying by $2$ — that's why the $\tfrac{1}{2}$ in the denominator turned into a $\times 2$.

Concept Check 🎯

A Setup Checklist

Every geometric-sum problem boils down to pinning three numbers, in this order:

1. $a_1$ — the first term you're adding.
2. $r$ — second term over first.
3. $n$ — how many terms the problem asks for.

Get those right and the formula does the rest. Let's set one up.

Set Up the Sum 🔽

You want the sum of the first 5 terms of $1,\,4,\,16,\,64,\dots$

Don't Fear the Negatives

When $r > 1$, both $1 - r^{n}$ (top) and $1 - r$ (bottom) come out negative — and the two negatives cancel, leaving a positive sum. That cancellation is built in, so trust the formula even when the intermediate fractions look ugly.

Compute the Sum 🧮

Use $S_n = a_1\cdot\dfrac{1-r^n}{1-r}$.

1) First 4 terms of $5,\,10,\,20,\,40,\dots$ $\Rightarrow S_4 = \,?$ 2) First 3 terms of $7,\,21,\,63,\dots$ $\Rightarrow S_3 = \,?$

🔁 Geometric Sequences

Part 5 of 5 — Applications & Mastery Check

You can now (1) find $r$, (2) write explicit and recursive rules, (3) find missing terms with the geometric mean, and (4) sum a finite series. Let's apply it to the real world and finish with a mastery check.

Real-World Growth & Decay

Geometric sequences model anything that grows or shrinks by a constant percent each period.

Multiplier rule: grow by $p\%$ → $r = 1 + \dfrac{p}{100}$; decay by $p\%$ → $r = 1 - \dfrac{p}{100}$.

Example: Bacteria triple every hour

A colony starts at $200$ cells and triples each hour. Since the starting count is at hour $0$, it's cleanest to track the count by hours elapsed $t$ rather than by term number — careful with the indexing!

Model the count $C$ at hour $t$ (with $t=0$ the start) as $C = 200 \cdot 3^{\,t}$. After $4$ hours:

$C = 200 \cdot 3^{4} = 200 \cdot 81 = 16{,}200 \text{ cells}$

💡 When a problem counts from hour $0$, use $a_1 \cdot r^{t}$ with exponent $t$. When it asks for the "$n$th term," use exponent $n-1$. Read carefully which one is wanted.

Application Check 🎯

Pull It All Together

The next drill mixes all four skills in one place — the explicit formula, the geometric mean, and the finite sum. Decide which tool each part needs before you compute:

• "Find the $n$th term" → $a_n = a_1 r^{\,n-1}$.
• "Find the term between two numbers" → $\sqrt{ab}$.
• "Add up the first $n$ terms" → $S_n = a_1\dfrac{1-r^n}{1-r}$.

Mixed Mastery Drill 🧮

1) $a_1 = 2,\; r = 3$. Find $a_6$. 2) Geometric mean of $8$ and $32$ $= \,?$ 3) Sum of the first $4$ terms of $1,\,3,\,9,\,27,\dots$ $\Rightarrow S_4 = \,?$

Quick Reference

⚠️ Top traps: the exponent is $n-1$ (not $n$); use the geometric mean $\sqrt{ab}$ (not the average); and $r$ can be negative or a fraction.

Exit Quiz ✅

Answer all three to finish the lesson.