What Is the Common Difference in the Following Arithmetic Sequence?

The common difference in an arithmetic sequence is the constant number you add to each term to get the next one. You find it by subtracting any term from the term immediately after it, as in d = a₂ − a₁. If that value is the same for every consecutive pair, the sequence is arithmetic and that number is the common difference. So when asked what is the common difference in the following arithmetic sequence, the answer is simply the fixed gap between adjacent terms, and one subtraction gives it to you.

What the Common Difference Means

An arithmetic sequence is an ordered list of numbers where the gap between one term and the next never changes. That fixed gap is the common difference, usually written as the letter d. Each term equals the previous term plus d.

For example, in 2, 5, 8, 11, each step adds 3, so d = 3. The same idea lets you build the sequence forward from any starting value: keep adding d and you generate the whole list. This constant step is what separates arithmetic sequences from other patterns such as geometric sequences, where you multiply by a fixed ratio instead.

Arithmetic progressions show up in more advanced number theory too. A group of primes that sit at equal intervals — say 3, 7, 11 — forms what is called an arithmetic progression of primes, and the shared gap is described as the common difference in that progression (Wikipedia). The core definition stays the same: one steady amount repeated between neighbors.

Positive, Negative, and Zero Differences

The common difference does not have to be a positive number:

• If d is positive, the sequence grows (4, 7, 10, 13).
• If d is negative, the sequence shrinks (20, 17, 14, 11).
• If d is zero, every term is identical (6, 6, 6, 6).

Recognizing the sign early tells you which direction the sequence is heading.

How to Find the Common Difference

The method is a single subtraction, applied to terms that are next to each other. Pick any term and subtract the one right before it:

[ d = a_{n} - a_{n-1} ]

In plain terms: take the second term minus the first, or the fifth minus the fourth — as long as the two terms are consecutive, the result is d. Khan Academy covers this term-by-term approach in its arithmetic sequences lessons (Khan Academy).

To be sure a sequence really is arithmetic, check more than one pair. Compute the gap for the first two terms, then for another neighboring pair. If both give the same number, the sequence is arithmetic and you have confirmed d. If the gaps differ, the sequence is not arithmetic at all.

A Quick Check You Can Do in Your Head

Subtract the smaller neighbor from the larger one when the sequence is increasing, and keep the sign in mind when it is decreasing. For a decreasing sequence like 15, 11, 7, the difference is 11 − 15 = −4, not +4. Getting the sign right matters because it tells you whether terms rise or fall.

Worked Example Step by Step

Take the sequence: 7, 12, 17, 22, 27.

Step 1 — Subtract the first pair. 12 − 7 = 5. So the candidate common difference is 5.

Step 2 — Test the next pair. 17 − 12 = 5. Still 5.

Step 3 — Confirm with another pair. 22 − 17 = 5, and 27 − 22 = 5.

Because every consecutive pair produces the same value, the common difference is d = 5. You can verify it by adding 5 back each time: 7 + 5 = 12, 12 + 5 = 17, and so on, which reproduces the original list.

One subtraction would have been enough to state the answer, but checking several pairs is what proves the pattern is genuinely arithmetic rather than a coincidence.

Common Mistakes to Avoid

Most errors come from a handful of small slips. Watch for these:

• Subtracting non-adjacent terms. Using 22 − 7 in the example gives 15, which is not the common difference — it is three steps’ worth of change. Always subtract terms that are directly next to each other.
• Sign errors. In a decreasing sequence, forgetting the negative sign flips your answer. Subtract in the same order every time (next term minus previous term) so the sign stays consistent.
• Confusing arithmetic with geometric sequences. Arithmetic sequences add a constant; geometric sequences multiply by a constant. In 3, 6, 12, 24 the pattern is multiplication by 2, so there is no common difference — that sequence is geometric, and its multiplier is called the common ratio (Purplemath).
• Assuming a pattern after one pair. Two terms always produce some difference, but a sequence only counts as arithmetic if the difference holds for every consecutive pair.

Spotting a Non-Arithmetic Sequence

Suppose you see 1, 4, 9, 16. The gaps are 3, 5, 7 — they change each time, so the sequence has no common difference. When the differences are not constant, stop and consider other patterns instead of forcing an arithmetic label.

Formulas That Use the Common Difference

Once you know d, you can jump straight to any term without listing everything in between. The explicit formula for an arithmetic sequence is:

[ a_n = a_1 + (n-1)d ]

Here a₁ is the first term and n is the position of the term you want. To find the 20th term of 7, 12, 17, 22, 27, plug in a₁ = 7 and d = 5: a₂₀ = 7 + (20 − 1)(5) = 7 + 95 = 102. You reach the same result by adding 5 nineteen times, but the formula saves the effort.

There is also a recursive formula, which states each term in terms of the one before it: aₙ = aₙ₋₁ + d, starting from a given first term. The recursive version shows how the sequence builds step by step, while the explicit version leaps directly to any position.

Recovering Missing Terms

Because d is constant, you can also work backward. If a sequence reads 9, 14, ___, 24, the gap between the known first pair is 5, and 24 − 14 = 10 spans two steps, confirming d = 5. The missing term is 14 + 5 = 19.

The Common Difference in Real Contexts

Arithmetic sequences describe any situation with a steady, equal change:

• Savings: depositing the same amount each month grows a balance by a fixed increment.
• Seating rows: an auditorium with 20 seats in the first row and 3 more in each following row uses d = 3.
• Temperature or elevation changes: a steady rise or fall per hour or per kilometer follows an arithmetic pattern.
• Schedules: evenly spaced time slots form an arithmetic sequence of minutes.

In each case, d is the repeated step, and recognizing it turns a word problem into a simple formula lookup.

Connecting d to the Sum

The common difference also appears in the formula for the sum of the first n terms of an arithmetic sequence:

[ S_n = \frac{n}{2}\big(2a_1 + (n-1)d\big) ]

For 7, 12, 17, 22, 27, the sum is 7 + 12 + 17 + 22 + 27 = 85. Using the formula with n = 5, a₁ = 7, and d = 5: S₅ = (5/2)(14 + 20) = (5/2)(34) = 85. The same constant step that defines the sequence also drives its total.

Key Takeaways

• The common difference (d) is the constant amount added between consecutive terms of an arithmetic sequence.
• Find it by subtracting any term from the term immediately after it: d = a₂ − a₁.
• Confirm the value using at least one more neighboring pair; if the gaps match, the sequence is arithmetic.
• Worked example: in 7, 12, 17, 22, 27, every step adds 5, so d = 5.
• d can be positive, negative, or zero, which makes the sequence grow, shrink, or stay flat.
• Avoid subtracting non-adjacent terms, dropping negative signs, and mixing up arithmetic (adding) with geometric (multiplying) sequences.