Skip to content

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

Sum of n Terms of an AP and Arithmetic Mean

Updated 10 October 2026 · Fact-checked

An AP is a list where each term is the previous term plus a fixed common difference d. Its sum of n terms is Sn = n/2 × [2a + (n − 1)d], or Sn = n/2 × (first term + last term). To insert arithmetic means between two numbers, find d first, then build the terms.

Understand Sum of n Terms of an AP and Arithmetic Mean

An arithmetic progression (AP) is a list of numbers in which the gap between neighbouring terms is always the same. That gap is the common difference (d). In 5, 8, 11, 14, the first term a is 5 and d is 3.

The sum of the first n terms is written Sn. You could add terms one by one, but that is slow. The formula works because pairs from both ends add up to the same value. In 5, 8, 11, 14, the pairs 5 + 14 and 8 + 11 both give 19. So the sum is the number of pairs times that pair total. With n terms there are n/2 pairs, so Sn = n/2 × (first + last).

If the last term is not given, replace it with a + (n − 1)d. This gives Sn = n/2 × [2a + (n − 1)d]. Use the first form when you know the last term. Use the second when you know d.

The arithmetic mean of two numbers x and y is (x + y) ÷ 2. If three numbers are in AP, the middle one is the arithmetic mean of the other two. This is the main property of an AP.

To insert n arithmetic means between two numbers a and b, you form an AP that starts at a and ends at b with n terms in between. That AP has n + 2 terms in total, so b = a + (n + 1)d. Solve for d, then write out the terms.

Key formulas to remember

Sum of n terms (using d)
Sn = n/2 × [2a + (n − 1)d]
Use when a, d and n are known.
Sum of n terms (using last term)
Sn = n/2 × (a + l)
Use when the first term a and last term l are known. Here l = a + (n − 1)d.
nth term from sums
Tn = Sn − S(n−1)
Valid for n ≥ 2. T1 = S1.
Arithmetic mean of two numbers
A = (x + y) ÷ 2
If x, A, y are in AP, then 2A = x + y.
Inserting n arithmetic means between a and b
d = (b − a) ÷ (n + 1)
The AP has n + 2 terms. The means are a + d, a + 2d, ..., a + nd.
Sum of the n inserted means
Sum of means = n × (a + b) ÷ 2
Equals n times the AM of a and b.
Three terms in AP
Take a − d, a, a + d
Their sum is 3a, so d cancels out when you add them.
Sum of first n natural numbers
1 + 2 + ... + n = n(n + 1) ÷ 2
A special case with a = 1 and d = 1.

How to solve Sum of n Terms of an AP and Arithmetic Mean questions

Use this method for any question on the sum of an AP or on arithmetic means.

  1. 1Check that the terms form an AP, meaning the difference between consecutive terms is constant.
  2. 2Write down what is given: a, d, n, last term l, or Sn. Mark what is asked.
  3. 3If the last term is unknown, find it with l = a + (n − 1)d. If n is unknown, find it from the last term first.
  4. 4Pick the sum formula that fits the data: n/2 × (a + l) if you know l, otherwise n/2 × [2a + (n − 1)d].
  5. 5For arithmetic means, find d = (b − a) ÷ (n + 1) and then add d repeatedly from a.
  6. 6Substitute carefully. Work out the bracket first, then multiply by n/2.
  7. 7If the question gives Sn and asks for n, form an equation and solve it. Reject negative or fractional n.
  8. 8Check your answer by testing the first term or the last term.

Quickest way: Pair-total shortcut

When to use it: Use it when the options are far apart and you can find the first and last terms quickly.

  1. Find the first term and last term.
  2. Add them to get the pair total.
  3. Multiply by n and divide by 2.
  4. For the AM of two numbers, just add them and halve.
  5. For inserting means, compute d once and list the terms. Check that the last mean plus d equals b.
  6. If three numbers are in AP, use 2 × middle = sum of the other two. This avoids solving for d.

Common mistakes in Sum of n Terms of an AP and Arithmetic Mean

  • Using d = (b − a) ÷ n when inserting n means.

    Students forget that the two end numbers are also terms, so there are n + 1 gaps.

    Fix: Always divide by (n + 1). Count the gaps between a and b, not the means.

  • Using the wrong n in Sn = n/2 × (a + l), for example by miscounting the terms.

    Students count the terms in the list by eye or take the last term as the number of terms, so the multiplier n/2 is wrong even when a and l are right.

    Fix: Find n from l = a + (n − 1)d first, then apply the sum formula.

  • Writing Sn = n/2 × [2a + nd].

    The (n − 1) is dropped from memory.

    Fix: Check with n = 1. The sum must equal a. This works only with (n − 1).

  • Accepting a negative or fractional value of n.

    When solving a quadratic for n, students keep both roots.

    Fix: n is a positive whole number. Discard any other root.

  • Giving the AM of the end numbers as the sum of the inserted means.

    Students find (a + b) ÷ 2 and stop, forgetting that it is the value of a single mean, not the total of all the inserted means.

    Fix: The AM of a and b is (a + b) ÷ 2. The sum of n inserted means is n × (a + b) ÷ 2.

Worked examples

Example 1

Find the sum of the first 20 terms of the AP 5, 9, 13, 17, ...

Show the solution
  1. Here a = 5, d = 9 − 5 = 4 and n = 20.
  2. Sn = n/2 × [2a + (n − 1)d].
  3. S20 = 20/2 × [2 × 5 + 19 × 4].
  4. = 10 × [10 + 76].
  5. = 10 × 86 = 860.

Answer: 860

Example 2

Insert 4 arithmetic means between 7 and 37. Find the sum of the four means.

Show the solution
  1. Here a = 7, b = 37 and n = 4 means, so the AP has 6 terms.
  2. d = (b − a) ÷ (n + 1) = (37 − 7) ÷ 5 = 6.
  3. The terms are 7, 13, 19, 25, 31, 37.
  4. The four means are 13, 19, 25, 31.
  5. Their sum is 13 + 19 + 25 + 31 = 88.
  6. Check: n × (a + b) ÷ 2 = 4 × 44 ÷ 2 = 88.

Answer: The means are 13, 19, 25 and 31. Their sum is 88.

Exam tips

  • Questions on this topic are usually one or two steps. Identify a, d and n first, then pick one formula.
  • When options include a value from d = (b − a) ÷ n, that is a trap option. Divide by n + 1.
  • If the problem gives the sum of three numbers in AP, take them as a − d, a, a + d. The sum gives a immediately.
  • There is no negative marking, so attempt every question. Use the check of testing the first term to eliminate wrong options.
  • Word problems on savings or instalments that rise by a fixed amount each period are AP sums. Identify the first amount and the fixed increase.

Practice questions from Arithmetic Progression and Geometric Progression

Sum of n Terms of an AP and Arithmetic Mean in other exams

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

Sum of n Terms of an AP and Arithmetic Mean: frequently asked questions

Which formula should I use for the sum of an AP?

Use Sn = n/2 × (a + l) when you know the first and last terms. Use Sn = n/2 × [2a + (n − 1)d] when you know the common difference instead. Both give the same answer.

How do I insert arithmetic means between two numbers?

Treat the two numbers as the first and last terms of an AP. If you need n means, the AP has n + 2 terms, so d = (b − a) ÷ (n + 1). Then add d repeatedly to the first number.

What is the arithmetic mean of two numbers in AP terms?

It is the single number that sits exactly between them in an AP. For x and y it equals (x + y) ÷ 2. If x, A, y are in AP, then A is the arithmetic mean.

Can the sum of an AP be found if only Sn is given?

Yes. The nth term is Tn = Sn − S(n−1) for n ≥ 2, and the first term is S1. From these you can find a and d.