Arithmetic Sequences: Adding the Same Amount Each Time

An arithmetic sequence is the simplest pattern of growth: add the same number, every time.

2, 5, 8, 11, 14, ...

Each term is 3 more than the one before. That constant gap—the common difference—defines the entire sequence. Know the first term and the difference, and you can find any term without computing the ones before it.

This is the pattern of steady, predictable growth. Not exponential explosion. Not random fluctuation. Just the same amount, added again and again.

The Formula That Jumps Ahead

Here's what makes arithmetic sequences powerful: you don't need to count your way to the 100th term.

If the first term is a₁ and the common difference is d, then the nth term is:

aₙ = a₁ + (n - 1)d

Why n - 1? Because to get from the first term to the nth term, you add d exactly n - 1 times.

• 1st term: a₁ (no additions yet)
• 2nd term: a₁ + d (one addition)
• 3rd term: a₁ + 2d (two additions)
• nth term: a₁ + (n-1)d

Let's test it. The sequence 2, 5, 8, 11, ... has a₁ = 2 and d = 3.

The 50th term? a₅₀ = 2 + (49)(3) = 2 + 147 = 149.

No need to write out 50 terms. The formula leaps straight to the answer.

Finding the Common Difference

Given any two consecutive terms, the common difference is just their gap:

d = aₙ₊₁ - aₙ

For the sequence 7, 12, 17, 22, ..., the difference is 12 - 7 = 5.

But you can also find d from any two terms if you know their positions:

d = (aₘ - aₙ) / (m - n)

If the 3rd term is 11 and the 7th term is 23, then:

d = (23 - 11) / (7 - 3) = 12 / 4 = 3

The terms are 4 positions apart, and the gap is 12, so each step is 3.

Arithmetic Sequences Are Linear

Plot an arithmetic sequence on a graph—term number on the x-axis, term value on the y-axis—and you get a straight line.

This isn't coincidence. The formula aₙ = a₁ + (n-1)d can be rewritten as:

aₙ = dn + (a₁ - d)

That's the equation of a line: y = mx + b, where the slope is d and the y-intercept is a₁ - d.

Arithmetic sequences are the discrete version of linear functions. They grow at a constant rate, just like a car traveling at steady speed. The first term tells you where you start. The common difference tells you how fast you're going.

The Middle Term Property

Here's a beautiful fact about arithmetic sequences: any term is the average of its neighbors.

Take 5, 9, 13. The middle term 9 is exactly the average of 5 and 13: (5 + 13) / 2 = 9.

This works for any three consecutive terms. If they're in arithmetic sequence, the middle one equals the average of the outer two.

This property is useful for checking your work and for finding missing terms. If you know the first and third terms of an arithmetic sequence, the second term is their average.

Negative Common Differences

Not all arithmetic sequences grow. Some shrink.

20, 17, 14, 11, 8, ...

Here d = -3. The sequence decreases by 3 each step.

The formula works the same way: aₙ = 20 + (n-1)(-3) = 20 - 3n + 3 = 23 - 3n.

At some point, the terms become negative:

• a₇ = 23 - 21 = 2
• a₈ = 23 - 24 = -1

An arithmetic sequence with negative d is a countdown. Eventually it passes through zero and keeps going into negative territory.

Finding Which Term Equals a Value

Sometimes you need to reverse the question: not "what is the 50th term?" but "which term equals 200?"

Set up the equation and solve:

aₙ = a₁ + (n-1)d = target value

For the sequence 7, 12, 17, 22, ... (a₁ = 7, d = 5), which term equals 152?

7 + (n-1)(5) = 152 (n-1)(5) = 145 n - 1 = 29 n = 30

The 30th term is 152.

If the answer isn't a whole number, the target value isn't in the sequence. The value 150, for instance, would give n = 29.6—not a valid position. 150 isn't a term of this sequence.

Arithmetic Means

The arithmetic mean between two numbers is the number that makes an arithmetic sequence with them.

What's the arithmetic mean between 10 and 20? The number that makes 10, ?, 20 an arithmetic sequence.

That's just the average: (10 + 20) / 2 = 15. The sequence 10, 15, 20 has common difference 5.

You can insert multiple arithmetic means too. To insert k arithmetic means between a and b:

1. The resulting sequence has k + 2 terms total
2. The common difference is d = (b - a) / (k + 1)

Insert 3 arithmetic means between 4 and 24:

• Total terms: 5
• Common difference: (24 - 4) / 4 = 5
• Sequence: 4, 9, 14, 19, 24

The inserted means are 9, 14, and 19.

Examples in the World

Arithmetic sequences model steady change:

Salary increases: A starting salary of $40,000 with $2,000 annual raises gives: 40000, 42000, 44000, ...

Depreciation: A car loses $3,000 in value each year: 25000, 22000, 19000, ...

Counting by intervals: Seats in a theater section: row 1 has 20 seats, each row adds 2 more: 20, 22, 24, 26, ...

Stacking: Each layer of a brick wall offsets by the same amount.

Any situation with constant change—not percentage change, but absolute change—follows an arithmetic pattern.

The Limit of an Arithmetic Sequence

What happens as n → ∞?

If d > 0, the sequence grows without bound: aₙ → +∞.

If d < 0, the sequence decreases without bound: aₙ → -∞.

If d = 0, the sequence is constant: every term equals a₁.

Arithmetic sequences never level off (unless d = 0). They march steadily toward positive or negative infinity.

This means arithmetic sequences don't converge—they don't approach a finite limit. We'll see later that this has important implications for their series.

Why Arithmetic Sequences Matter

Arithmetic sequences are foundational because:

1. They're the simplest growth pattern. Before understanding exponential growth, you need to understand constant growth.
2. Their sums have a beautiful formula. Adding up an arithmetic sequence—called an arithmetic series—has a closed form we'll explore soon.
3. They connect to linear algebra. Arithmetic sequences are discrete samples of linear functions. The theory of lines and the theory of arithmetic sequences are the same theory.
4. They're everywhere. Whenever something changes by a fixed amount per step—not a fixed percentage, a fixed amount—you have an arithmetic sequence.

The arithmetic sequence says: growth doesn't have to accelerate. Sometimes adding the same amount, over and over, is exactly what the pattern needs.

Part 2 of the Sequences Series series.

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