Mathematics › 🌆 Grade 9 › Arithmetic & Geometric Sequences

Arithmetic & Geometric Sequences

Add the same amount every step and you get a straight line of dots; multiply by the same factor and you get a curve that explodes.

Grade 9Algebra 1

Arithmetic & Geometric Sequences — illustration

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The big idea: A sequence is a function whose inputs are just the counting numbers 1, 2, 3, … — which makes it discrete, a list of separate points rather than a smooth curve. An arithmetic sequence adds the same fixed amount every step, producing linear growth; a geometric sequence multiplies by the same fixed factor every step, producing exponential growth. Recognising which pattern you are looking at — constant difference or constant ratio — tells you everything about how the sequence behaves far down the line.

🎯 By the end, you'll be able to

Explain why a sequence is a discrete function defined only on the positive integers
Write the explicit and recursive formulas for an arithmetic sequence given a₁ and d
Write the explicit and recursive formulas for a geometric sequence given a₁ and r
Compute a partial sum of an arithmetic or geometric sequence and state when an infinite geometric sum exists

📎 You should already know

Function notation (f(x))
Exponents with whole-number powers

A function on the counting numbers

A sequence is a list of numbers in order: a₁, a₂, a₃, … It is really a function — feed in a position n, get out a term a_n — except its domain is restricted to the positive integers 1, 2, 3, … That restriction makes a sequence discrete: there is no a_2.5, so a sequence is always plotted as separate dots, never connected by a smooth line.

Two patterns of change show up constantly: adding the same amount each step, and multiplying by the same amount each step. They look similar at first glance but behave completely differently as n grows.

🔑 Arithmetic: constant difference. Geometric: constant ratio.

In an arithmetic sequence, each term is the previous term plus a fixed common difference d. In a geometric sequence, each term is the previous term times a fixed common ratio r. Spot which one you have by subtracting consecutive terms (constant result → arithmetic) or dividing them (constant result → geometric).

\[ \text{Arithmetic: } a_n = a_1 + (n-1)d \qquad a_n = a_{n-1} + d \]

Explicit formula (jump straight to term n) and recursive formula (build from the previous term), both seeded at a₁.

\[ \text{Geometric: } a_n = a_1 \cdot r^{\,n-1} \qquad a_n = r \cdot a_{n-1} \]

Same explicit-vs-recursive split, but multiplication replaces addition throughout.

Explicit vs. recursive

The explicit formula computes any term directly from n, without needing any earlier term — useful for jumping straight to, say, a₁₀₀. The recursive formula defines a term only in relation to the one before it, so it must always be paired with a starting value, the seed a₁; without that seed, a recursive rule describes a pattern of change but not a single actual sequence.

🎮 Sequence Builder LIVE

Set a₁, d or r, and n to watch the discrete points, the explicit formula, and the running partial sum update together.

📝 Worked example: An arithmetic sequence starts at a₁ = 4 with common difference d = 3. Find a₈.

Use the explicit formula \( a_n = a_1 + (n-1)d \) with n = 8.
Substitute: \( a_8 = 4 + (8-1)\times 3 = 4 + 7\times 3 \).
Compute: \( 4 + 21 = 25 \).

✓ \( a_8 = \mathbf{25} \)

📝 Worked example: A geometric sequence starts at a₁ = 5 with common ratio r = 2. Find a₆ and the partial sum S₆.

Explicit formula: \( a_n = a_1 \cdot r^{\,n-1} \), so \( a_6 = 5\times 2^{5} = 5\times 32 = 160 \).
Since r = 2 ≠ 1, use \( S_n = \dfrac{a_1(1-r^n)}{1-r} \) with n = 6.
\( S_6 = \dfrac{5(1-2^6)}{1-2} = \dfrac{5(1-64)}{-1} = \dfrac{5(-63)}{-1} \).

✓ \( a_6 = \mathbf{160} \), and \( S_6 = \mathbf{315} \).

Two very different growth patterns

Because an arithmetic sequence adds the same amount repeatedly, its terms grow linearly — plot a_n against n and the dots line up on a straight line. A geometric sequence, multiplying repeatedly, grows (or shrinks) exponentially — the dots start out looking almost flat and then bend sharply upward, the same “knee” shape you saw with exponential growth models. That connection is not a coincidence: a geometric sequence is exactly the discrete, term-by-term version of continuous exponential growth.

⚠️ It's (n − 1), not n

Both explicit formulas use (n − 1) as the exponent or multiplier on d or r, never plain n. Check it at n = 1: a₁ = a₁ + (1−1)d = a₁ + 0 = a₁, exactly as it should be. Using n instead of (n−1) shifts every single term by one position and silently breaks the whole sequence.

⚠️ A recursive rule needs its seed, and dots are not a curve

A recursive rule like a_n = a_n−1 + 3 is meaningless on its own — without a stated a₁, it could describe infinitely many different sequences. And however smooth a sequence’s growth looks, it is only ever defined at whole-number positions; joining the dots with a curve misrepresents it as a continuous function it is not.

✨ When does the geometric sum settle down?

The finite partial sum S_n = a₁(1 − rⁿ)÷(1 − r) requires r ≠ 1 (when r = 1 every term is identical, so S_n = n·a₁ directly). Extending the sum to infinitely many terms only produces a finite value, a₁÷(1 − r), when |r| < 1 — each new term must be shrinking toward zero, or the running total never settles.

Check your understanding

1. Why is a sequence called 'discrete'?

A sequence's domain is restricted to the positive integers 1, 2, 3, …, so there is no term 'between' a₁ and a₂ — that restriction is what makes it discrete.

2. A sequence begins 5, 5, 5, 5… (constant ratio r = 1) vs. 5, 8, 11, 14… What identifies the second sequence as arithmetic?

8−5 = 3, 11−8 = 3, 14−11 = 3: a constant difference between consecutive terms is the defining test for an arithmetic sequence.

3. For an arithmetic sequence with a₁ = 10 and d = −2, what is a₅?

a₅ = a₁ + (5−1)d = 10 + 4×(−2) = 10 − 8 = 2.

4. Why can't the recursive rule aₙ = aₙ₋₁ × 3 alone define a specific sequence?

A recursive rule only describes how to get the next term from the previous one — without a seed value a₁, infinitely many different sequences satisfy the same rule.

5. The infinite geometric sum a₁/(1 − r) gives a finite value only when:

Each added term must shrink toward zero for the running total to settle on a finite value, which happens only when |r| < 1.

✅ Key takeaways

A sequence is a discrete function defined only at positive-integer positions n — never join its terms with a smooth curve.
Arithmetic: constant common difference d, explicit aₙ = a₁ + (n − 1)d, recursive aₙ = aₙ₋₁ + d seeded at a₁.
Geometric: constant common ratio r, explicit aₙ = a₁·r^(n−1), recursive aₙ = r·aₙ₋₁ seeded at a₁.
Arithmetic sequences grow linearly; geometric sequences grow exponentially — the same 'knee' shape seen in exponential growth.
The explicit formula always uses (n − 1), not n; a geometric infinite sum exists only when |r| < 1.