Arithmetic and Geometric Sequences

🔢 Arithmetic & Geometric Sequences

Part 1 of 5 — Sequences & the Common Difference

Topics in This Part

🔑 Key Concept: A sequence is an ordered list of numbers. The numbers are called terms. The whole lesson is about spotting the pattern that gets you from one term to the next — and writing a rule for it.

What Is a Sequence?

A sequence is just a list of numbers in a set order:

$2,\; 5,\; 8,\; 11,\; 14,\; \dots$

We name the terms with subscripts:

The little number is the position (or index). So $a_4 = 11$ means "the 4th term is $11$."

💡 The "$\dots$" means the pattern keeps going forever. A sequence can be finite (it stops) or infinite (it never ends).

Arithmetic Sequences

In an arithmetic sequence, you get the next term by adding the same number every time. That fixed number is the common difference, written $d$.

$2,\; 5,\; 8,\; 11,\; 14,\; \dots \qquad d = +3$

To find $d$, subtract any term from the one after it:

$d = a_2 - a_1 = 5 - 2 = 3$

It must be the same gap every time, or the sequence is not arithmetic.

🔑 Key Idea: Arithmetic = repeated addition of a constant $d$. If $d$ is negative, the terms go down.

Concept Check 🎯

Find the Common Difference 🧮

For each arithmetic sequence, find $d$ by computing $a_2 - a_1$. (Negative answers are fine.)

1) $6, 10, 14, 18, \dots \quad d = \,?$ 2) $30, 24, 18, 12, \dots \quad d = \,?$ 3) $-7, -4, -1, 2, \dots \quad d = \,?$

Predicting the Next Term

Once you know $d$, the next term is just current term $+\, d$:

$a_{n+1} = a_n + d$

This is called a recursive rule — it tells you how to get the next term from the one you have.

Example: $4, 11, 18, 25, \dots$ has $d = 7$. The next term is $25 + 7 = 32$.

⚠️ Recursion is great for the next term, but slow for far-away terms. To jump straight to the 100th term without listing all of them, we need a direct formula — that's Part 2.

Extend the Pattern 🔽

The arithmetic sequence is $4, 11, 18, 25, \dots$. Fill in each blank using $a_{n+1} = a_n + d$.

🔢 Arithmetic & Geometric Sequences

Part 2 of 5 — The Explicit Formula for Arithmetic Sequences

🔑 The Idea: Instead of adding $d$ over and over, the explicit formula lets you plug in a position $n$ and get the term $a_n$ directly — even the 500th term.

The Explicit (nth-Term) Formula

For an arithmetic sequence with first term $a_1$ and common difference $d$:

$\boxed{\,a_n = a_1 + (n - 1)\,d\,}$

Why $(n-1)$? To reach the $n$th term, you start at $a_1$ and add $d$ a total of $(n-1)$ times — because there are $n-1$ gaps between $n$ terms.

💡 The number of steps is always one less than the term number. That single "$-1$" is the most common place students slip.

Worked Example

Find the 10th term of $3, 7, 11, 15, \dots$

First, identify the pieces: $a_1 = 3$ and $d = 7 - 3 = 4$.

Plug into $a_n = a_1 + (n-1)d$ with $n = 10$:

$a_{10} = 3 + (10 - 1)(4) = 3 + (9)(4) = 3 + 36 = 39$

✅ Check: Count up — $3, 7, 11, 15, 19, 23, 27, 31, 35, 39$. The 10th term is indeed $39$. ✓

Building the Rule

You can also write the whole formula for this sequence by simplifying:

$a_n = 3 + (n-1)(4) = 3 + 4n - 4 = 4n - 1$

Now any term is one step away: $a_{20} = 4(20) - 1 = 79$.

Build the Formula 🔽

You're writing the explicit formula for the arithmetic sequence $5, 8, 11, 14, \dots$. Fill in each piece.

Find the Term 🧮

Use $a_n = a_1 + (n-1)d$ for each.

1) $a_1 = 2,\; d = 5$. Find $a_8 = \,?$ 2) $4, 9, 14, 19, \dots$ Find the 12th term, $a_{12} = \,?$ 3) $a_1 = 50,\; d = -3$. Find $a_{10} = \,?$

Working Backwards 🎯

🔢 Arithmetic & Geometric Sequences

Part 3 of 5 — Geometric Sequences & the Common Ratio

🔑 New Pattern: Arithmetic sequences add the same number. Geometric sequences multiply by the same number every time. That multiplier is the common ratio, $r$.

Geometric Sequences

In a geometric sequence, each term is the previous term times a fixed number $r$:

$2,\; 6,\; 18,\; 54,\; 162,\; \dots \qquad r = \times 3$

To find the common ratio, divide any term by the one before it:

$r = \frac{a_2}{a_1} = \frac{6}{2} = 3$

It must be the same ratio every step:

🔑 Key Idea: Geometric = repeated multiplication by a constant $r$. If $0 < r < 1$, the terms shrink; if $r > 1$, they grow fast.

Find the Common Ratio 🧮

Find $r$ by computing $\dfrac{a_2}{a_1}$. (Fractions like $1/2$ are fine.)

1) $3, 12, 48, 192, \dots \quad r = \,?$ 2) $7, 14, 28, 56, \dots \quad r = \,?$ 3) $80, 40, 20, 10, \dots \quad r = \,?$ (fraction or decimal)

The Explicit Formula for Geometric Sequences

For a geometric sequence with first term $a_1$ and common ratio $r$:

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

Just like arithmetic, the exponent is $(n-1)$ — because reaching the $n$th term takes $n-1$ multiplications.

Worked Example

Find the 5th term of $2, 6, 18, \dots$

Here $a_1 = 2$ and $r = \frac{6}{2} = 3$. With $n = 5$:

$a_5 = 2 \cdot 3^{\,5-1} = 2 \cdot 3^4 = 2 \cdot 81 = 162$

✅ Check: $2, 6, 18, 54, 162$ — the 5th term is $162$. ✓

Concept Check 🎯

Find the Term 🧮

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

1) $a_1 = 3,\; r = 2$. Find $a_6 = \,?$ 2) $1, 5, 25, 125, \dots$ Find $a_4 = \,?$ 3) $a_1 = 4,\; r = 3$. Find $a_4 = \,?$

🔢 Arithmetic & Geometric Sequences

Part 4 of 5 — Telling Them Apart & Real-World Models

🔑 The Big Question: Given a sequence, is it arithmetic, geometric, or neither? Then: how do sequences model real situations like savings, populations, and bouncing balls?

Arithmetic vs. Geometric

The fastest test: look at how you get from one term to the next.

💡 Quick test: Compute both the difference and the ratio of the first two pairs. If the difference is constant → arithmetic. If the ratio is constant → geometric. If neither is constant → it's neither.

Classify Each Sequence 🔽

Decide whether each sequence is arithmetic, geometric, or neither.

Sequences in the Real World

Arithmetic models describe steady, repeated adding:

You save $50 in a jar, then add $15 every week. Week $n$ has $a_n = 50 + (n-1)(15)$ dollars.

Geometric models describe repeated multiplying (percent growth/decay):

A bacteria colony of $200$ doubles every hour. After $n$ hours there are $a_n = 200 \cdot 2^{\,n}$ cells. (Here we use the exponent $n$ because "after 1 hour" already means one doubling.)

⚠️ Watch the starting point. "Doubles every hour starting at 200" can be written as a sequence $200, 400, 800, \dots$ where $a_1 = 200, r = 2$, giving $a_n = 200\cdot 2^{n-1}$ for the $n$th term in the list. Read carefully whether $n$ counts terms or hours.

Application Check 🎯

Model It 🧮

1) You start with $50 and add $15 each week: $a_n = 50 + (n-1)(15)$. How many dollars after week 6 ($a_6$)? $\,?$ 2) A colony starts at $200$ and doubles each hour: the list is $200, 400, 800, \dots$ with $a_n = 200\cdot 2^{\,n-1}$. How many cells is the 4th term ($a_4$)? $\,?$

🔢 Arithmetic & Geometric Sequences

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) find $d$ and $r$, (2) use both explicit formulas, (3) tell the two types apart, and (4) model real situations. Let's put it all together.

Quick Reference

⚠️ Top traps: use $(n-1)$, not $n$, in both formulas; subtract for $d$ but divide for $r$; and for geometric terms, apply the exponent before multiplying by $a_1$.

Mixed Practice 🔽

Pull the right tool for each problem.

Mixed Drill 🧮

1) Arithmetic: $a_1 = 9,\; d = 6$. Find $a_7 = \,?$ 2) Geometric: $a_1 = 2,\; r = 5$. Find $a_4 = \,?$ 3) Arithmetic sequence $11, 8, 5, 2, \dots$ Find $a_9 = \,?$

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.