Arithmetic Sequences, Common Differences, and nth-Term Rules

A sequence is an ordered list of numbers that follows a rule. An arithmetic sequence changes by the same amount from one term to the next. That fixed amount is the common difference, d. For 4, 9, 14, 19, …, subtract consecutive terms: 9 − 4 = 5, 14 − 9 = 5, and 19 − 14 = 5. The common difference is 5.

Find the common difference

Subtract a term from the term immediately after it: d = a₂ − a₁, or generally d = aₙ − aₙ₋₁. Keep the order consistent. A sequence such as 18, 13, 8, 3, … has d = 13 − 18 = −5. A negative common difference means the terms decrease by 5 each step; it does not mean the terms themselves must be negative.

If the difference is not constant, the sequence is not arithmetic. For example, 2, 4, 8, 16 multiplies by 2 each time, so it is geometric rather than arithmetic. Compare differences before assuming a pattern from only the first two terms.

Write a recursive rule

A recursive rule gives the first term and tells how to get each next term from the one before it. For 4, 9, 14, 19, …, state a₁ = 4 and aₙ = aₙ₋₁ + 5 for n ≥ 2. This rule is useful when generating terms in order, but it requires knowing the previous term to calculate a distant one.

Write an explicit nth-term rule

An explicit rule calculates any term directly: aₙ = a₁ + (n − 1)d. The factor n − 1 appears because the first term has zero steps after itself, the second term has one added difference, and the third term has two. For the sequence 4, 9, 14, 19, …, aₙ = 4 + 5(n − 1). The twentieth term is 4 + 5(19) = 99.

Check the rule at n = 1: a₁ = 4 + 5(0) = 4, the listed first term. This is a quick way to catch a common off-by-one error. Using a₁ + nd would give 9 for n = 1 and shift every term one place.

Find a term when the first term is not provided

If you know a later term and the common difference, use the same rule and solve for a₁ or n. Suppose a sequence has common difference 3 and a₁₀ = 32. Then 32 = a₁ + 9(3), so a₁ = 5. To find the position of 50 in the resulting sequence, solve 50 = 5 + (n − 1)3. This gives 45 = 3(n − 1), n − 1 = 15, and n = 16.

Connect a sequence to a linear relationship

The explicit rule aₙ = a₁ + (n − 1)d is linear in n. Each time n increases by 1, the term changes by d. A table of term number n and value aₙ forms points on a line, with slope d. The first term a₁ is the value at n = 1, not necessarily the y-intercept of the graph when the horizontal axis is n.

You can rewrite the formula as aₙ = dn + (a₁ − d). For 4, 9, 14, …, this is aₙ = 5n − 1. The coefficient 5 is the change per step. The constant −1 gives the value the linear expression would have at n = 0, a position before the sequence begins.

Model a fixed amount added each period

Suppose a savings jar starts with $18 and receives $7 every week. The sequence of balances beginning with the starting amount is 18, 25, 32, 39, …, with a₁ = 18 and d = 7. The twelfth listed balance is a₁₂ = 18 + (12 − 1)(7) = $95, which is the balance after 11 weekly deposits. After 12 deposits, the balance is $18 + 12($7) = $102, or the thirteenth listed value. State whether a problem counts the initial amount as a term or asks for deposits after the start.

The context helps identify a fixed difference. Adding the same number of dollars, seats, or pages each period is additive change. A fixed percentage increase usually produces a geometric sequence because the amount added depends on the current amount. Distinguish 'increase by 7 each week' from 'increase by 7 percent each week.'

If the first and later terms are known, you can find d from d = (aₙ − a₁)/(n − 1). For instance, if a₁ = 8 and a₆ = 33, then d = (33 − 8)/5 = 5. Verify by stepping five differences from 8 to 33. The index gap, not the later term number by itself, determines how many differences separate the values.

Common errors

• Subtracting terms in inconsistent order and switching the sign of d.
• Using n instead of n − 1 in the explicit nth-term formula.
• Calling a sequence arithmetic after checking only one difference.
• Confusing a constant difference with a constant ratio.
• Starting to count terms at zero when the problem labels the first term as a₁.
• Rounding an exact term or index before the calculation is complete.

A strong check is to compute two terms from the explicit rule and compare them with the pattern. The first should equal a₁, and the difference between consecutive calculated terms should equal d. If a proposed position is not a whole number, the requested value may not occur in the sequence.

Exam takeaway

Check consecutive differences to identify an arithmetic sequence. Find d by subtracting consecutive terms, then use aₙ = a₁ + (n − 1)d for a direct calculation. Substitute n = 1 to verify the rule and preserve the term numbering stated in the problem.

Apply it to the evidence or sentence

An arithmetic sequence changes by the same common difference d between consecutive terms. Find it by subtracting a term from the next: in 11, 7, 3, −1, d = −4. The explicit rule aₙ = a₁ + (n − 1)d gives the nth term without listing every prior value. Here a₁ = 11, so aₙ = 11 − 4(n − 1), and a₆ = −9. The factor n − 1 appears because the first term has zero steps from itself. Check the rule against at least two terms and distinguish the common difference from the term number.