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Fundamentals of Business Mathematics and Statistics · Arithmetic Progression and Geometric Progression

Arithmetic Progression: nth Term and Common Difference

Updated 10 October 2026 · Fact-checked

An arithmetic progression (AP) is a sequence in which each term differs from the previous one by a fixed number, the common difference d. The nth term is Tn = a + (n − 1)d, where a is the first term. Use it to find any term, d, or the number of terms.

Understand Arithmetic Progression: nth Term and Common Difference

A sequence is a list of numbers in a fixed order. An arithmetic progression (AP) is a sequence where you get each term by adding the same number to the term before it. Example: 5, 8, 11, 14, ... Each term is 3 more than the last.

That fixed number is the common difference, written d. You find it by subtracting any term from the term after it: d = second term − first term. If d is positive, the AP rises. If d is negative, it falls. If d is 0, every term is the same.

The first term is called a. To reach the 2nd term you add d once. To reach the 3rd term you add d twice. So to reach the nth term you add d exactly (n − 1) times. That is why Tn = a + (n − 1)d. The multiplier is (n − 1), not n.

Most exam questions give you two pieces of information and ask for a third. For example, you may be given two terms and asked for a and d, or given a, d and the last term and asked how many terms there are. Each case reduces to the same formula and, at most, two simple equations.

Key formulas to remember

Common difference
d = T2 − T1 = T3 − T2 = Tn − T(n−1)
Must be the same for every consecutive pair, or the sequence is not an AP.
nth term
Tn = a + (n − 1)d
a is the first term, n is the position of the term.
Number of terms
n = (l − a) ÷ d + 1
l is the last term. The answer must be a positive whole number.
Difference between any two terms
Tm − Tn = (m − n)d
Fastest way to find d when two terms are given.
Three terms in AP
2b = a + c
If a, b, c are in AP, the middle term is the average of the other two.

How to solve Arithmetic Progression: nth Term and Common Difference questions

Use this method for any AP question on the nth term, common difference or number of terms.

  1. 1Write down what is given: a, d, n, Tn or the values of particular terms.
  2. 2Write down what you must find.
  3. 3Check the sequence is an AP by confirming the difference between consecutive terms is constant.
  4. 4Write the formula Tn = a + (n − 1)d and substitute the known values.
  5. 5If two terms are given, form two equations, one for each term, and subtract them to get d first.
  6. 6Solve for the unknown, then find the other unknown by substitution if needed.
  7. 7For the number of terms, check that n is a positive whole number. If not, recheck your working.
  8. 8Put the answer back in the formula to verify it.

Quickest way: Subtract-and-divide shortcut

When to use it: When two terms are given, or when you need the number of terms between a first and last value.

  1. For two terms Tm and Tn, use d = (Tm − Tn) ÷ (m − n). No equations needed.
  2. Then find any term by moving from a known term: Tk = Tm + (k − m)d.
  3. For the number of terms, use n = (l − a) ÷ d + 1.
  4. If you need a term and the options are close, test one option with the formula instead of solving fully.

Common mistakes in Arithmetic Progression: nth Term and Common Difference

  • Using Tn = a + nd instead of a + (n − 1)d.

    You think each of the n terms adds d, but the first term has no d added.

    Fix: Check with n = 1. The formula must give T1 = a. Only (n − 1)d does this.

  • Finding d as first term − second term.

    You subtract in the wrong order.

    Fix: Always do later term − earlier term. For 20, 17, 14, d = 17 − 20 = −3.

  • Forgetting the + 1 when finding the number of terms.

    You stop after dividing (l − a) by d, which only counts the gaps.

    Fix: Number of terms = gaps + 1. For 3, 5, 7, 9: (9 − 3) ÷ 2 = 3 gaps, so 4 terms.

  • Dropping the negative sign of d in a decreasing AP.

    You treat d as a plain length and ignore direction.

    Fix: Keep the sign in every substitution. Write d = −3 in brackets while calculating.

  • Assuming a sequence is an AP after checking only one difference.

    You compare just the first two terms.

    Fix: Check at least two pairs. The sequence 2, 4, 8 has a first difference 2 but the next difference is 4, so it is not an AP.

  • Accepting a fractional or negative n.

    You do not check whether the answer makes sense.

    Fix: The number of terms must be a positive whole number. If not, recheck the arithmetic.

Worked examples

Example 1

The 4th term of an AP is 17 and the 9th term is 42. Find the first term, the common difference and the 15th term.

Show the solution
  1. Use Tm − Tn = (m − n)d with m = 9, n = 4: 42 − 17 = 5d.
  2. 25 = 5d, so d = 5.
  3. T4 = a + 3d = 17, so a + 15 = 17, giving a = 2.
  4. T15 = a + 14d = 2 + 14 × 5 = 2 + 70 = 72.
  5. Check: T9 = 2 + 8 × 5 = 42, which matches.

Answer: a = 2, d = 5, T15 = 72.

Example 2

How many terms are there in the AP 7, 13, 19, ..., 151? Also find which term equals 85.

Show the solution
  1. a = 7 and d = 13 − 7 = 6.
  2. Number of terms: n = (151 − 7) ÷ 6 + 1 = 144 ÷ 6 + 1 = 24 + 1 = 25.
  3. For the term equal to 85: 7 + (n − 1) × 6 = 85.
  4. (n − 1) × 6 = 78, so n − 1 = 13 and n = 14.
  5. Check: T14 = 7 + 13 × 6 = 85.

Answer: There are 25 terms, and 85 is the 14th term.

Exam tips

  • Questions are objective, so use d = (Tm − Tn) ÷ (m − n) to skip the two-equation method and save time.
  • Test the options in the formula when the question asks for a particular term or n. It is often faster than full algebra.
  • Watch for the wording 'which term' versus 'what is the term'. One asks for n, the other for Tn.
  • For three numbers in AP, use the middle-term idea 2b = a + c to form one quick equation.
  • Always check that n is a positive whole number before marking an option.

Practice questions from Arithmetic Progression and Geometric Progression

Arithmetic Progression: nth Term and Common Difference in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Arithmetic Progression: nth Term and Common Difference: frequently asked questions

How do you find the common difference in an AP?

Subtract any term from the term that follows it: d = T2 − T1. If two non-adjacent terms are given, use d = (Tm − Tn) ÷ (m − n). The value can be positive, negative or zero.

What is the nth term formula of an AP?

Tn = a + (n − 1)d, where a is the first term and d is the common difference. It gives the term at position n. Check it by putting n = 1, which should give a.

How do you find the number of terms in an AP?

Use n = (l − a) ÷ d + 1, where l is the last term. Divide the total gap by d, then add 1. The result must be a positive whole number.

Can the common difference be negative or zero?

Yes. A negative d gives a decreasing AP such as 30, 25, 20. A zero d gives a constant sequence such as 6, 6, 6, which is still an AP.