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Quantitative Aptitude · Sequence and Series

Arithmetic Mean and Insertion of Means: CA Foundation Quantitative Aptitude

Updated 1 October 2026 · Fact-checked

The arithmetic mean of two numbers a and b is (a + b) ÷ 2, the middle term of an AP a, A, b. To insert n arithmetic means between a and b, find the common difference d = (b − a) ÷ (n + 1), then add d repeatedly starting from a.

Understand Arithmetic Mean and Insertion of Means

An arithmetic mean (AM) of two numbers a and b is the number A that makes a, A, b an arithmetic progression. In an AP, each term exceeds the previous one by the same amount. So A − a = b − A, which gives A = (a + b) ÷ 2. It is simply the average of the two numbers.

Now suppose you want more than one number between a and b. If you place n numbers between them so that the whole list a, A₁, A₂, ..., Aₙ, b is an AP, those n numbers are called n arithmetic means between a and b.

The full list has n + 2 terms. The first term is a and the last term is b, which is the (n + 2)th term. Using the nth term formula of an AP, b = a + (n + 1)d. The list has n + 1 equal gaps, which is why d = (b − a) ÷ (n + 1).

Once you know d, the means are A₁ = a + d, A₂ = a + 2d, and so on up to Aₙ = a + nd. Their sum has a neat form: the sum of the n means equals n times the single AM of a and b, that is n(a + b) ÷ 2.

Key formulas to remember

AM of two numbers
A = (a + b) ÷ 2
a, A, b are in AP. Works for any two real numbers.
Common difference after inserting n means
d = (b − a) ÷ (n + 1)
There are n + 1 equal gaps, not n.
The k-th inserted mean
Aₖ = a + k·d, for k = 1, 2, ..., n
A₁ = a + d is the first mean and Aₙ = b − d is the last.
Sum of the n inserted means
A₁ + A₂ + ... + Aₙ = n(a + b) ÷ 2
Equals n times the AM of a and b. Does not include a and b.
Three terms in AP
2B = A + C
Use this to test whether a middle term is the AM of its neighbours.

How to solve Arithmetic Mean and Insertion of Means questions

Use this method for any question on the AM of two numbers or on inserting means.

  1. 1Identify the first number a and the last number b. Note how many means n are to be inserted.
  2. 2If only one mean is needed, compute (a + b) ÷ 2 and stop.
  3. 3For n means, compute d = (b − a) ÷ (n + 1). Keep the sign: d is negative if b is less than a.
  4. 4Write the means as a + d, a + 2d, ..., a + nd.
  5. 5If asked for a particular mean, use a + kd directly. Do not list all of them.
  6. 6If asked for the sum of the means, use n(a + b) ÷ 2.
  7. 7Check: the last mean plus d should equal b.

Quickest way: Gap counting and option elimination

When to use it: Use in MCQs where you must find d, a specific mean, or the sum of the means.

  1. Count gaps: n means give n + 1 gaps. Divide the total difference (b − a) by this count.
  2. For a specific mean, jump straight to a + kd without writing the others.
  3. For the sum of the means, use n × (a + b) ÷ 2. This takes seconds and needs no d.
  4. Test options: if a mean is a + kd, the next option value must differ from it by d.
  5. Check the last mean: b − d. If it does not match, you miscounted gaps.
  6. If d is a messy fraction and the options are all integers, recheck n + 1.

Common mistakes in Arithmetic Mean and Insertion of Means

  • Using d = (b − a) ÷ n instead of (b − a) ÷ (n + 1).

    Students count the means and forget that a and b are also terms, so the gaps are one more than the means.

    Fix: Always say: n means, n + 2 terms, n + 1 gaps.

  • Including a and b in the sum of the inserted means.

    The word 'series' makes students add the whole AP.

    Fix: The n means exclude a and b. Use n(a + b) ÷ 2, or subtract a + b from the total AP sum.

  • Losing the negative sign of d when b < a.

    Students subtract the smaller from the larger by habit.

    Fix: Always compute b − a in that order. A decreasing AP has a negative d.

  • Taking the first mean as a instead of a + d.

    Confusing the first term of the AP with the first inserted mean.

    Fix: The first mean is a + d. The term a itself is not a mean.

  • Taking the AM of two numbers as their product or GM.

    Mixing up AM with GM, which is √(ab).

    Fix: AM adds and halves. GM multiplies and takes the square root.

Worked examples

Example 1

Insert 4 arithmetic means between 3 and 28. The third mean is: (a) 18 (b) 20 (c) 23 (d) 13

Show the solution
  1. a = 3, b = 28, n = 4.
  2. d = (28 − 3) ÷ (4 + 1) = 25 ÷ 5 = 5.
  3. The means are 8, 13, 18, 23.
  4. The third mean is a + 3d = 3 + 15 = 18.
  5. Check: 23 + 5 = 28 = b.

Answer: (a) 18

Example 2

The sum of 5 arithmetic means between 11 and 41 is: (a) 130 (b) 150 (c) 156 (d) 260

Show the solution
  1. a = 11, b = 41, n = 5.
  2. Sum of the means = n(a + b) ÷ 2 = 5 × 52 ÷ 2.
  3. = 5 × 26 = 130.
  4. Check: d = 30 ÷ 6 = 5, means are 16, 21, 26, 31, 36. Sum = 130.

Answer: (a) 130

Example 3

If n arithmetic means are inserted between 2 and 38 and the common difference is 4, the value of n is: (a) 7 (b) 8 (c) 9 (d) 10

Show the solution
  1. d = (b − a) ÷ (n + 1), so 4 = 36 ÷ (n + 1).
  2. n + 1 = 36 ÷ 4 = 9.
  3. n = 8.
  4. Check: the AP has 10 terms, 2 + 9 × 4 = 38.

Answer: (b) 8

Exam tips

  • Questions are usually one-step: find d, find a specific mean, or find n. Do them mentally if you can.
  • Remember n + 1 gaps. Most wrong options are built from the n-gap error.
  • For the sum of the means, use n(a + b) ÷ 2 and skip finding d.
  • Always run the quick check: last mean + d = b. It takes five seconds and protects against 0.25 negative marking.
  • If the numbers look unfriendly and you are unsure, mark the question and return later.

Practice questions from Sequence and Series

Arithmetic Mean and Insertion of Means: frequently asked questions

What is the arithmetic mean between two numbers in CA Foundation?

It is the number A such that a, A, b are in AP. It equals (a + b) ÷ 2, the simple average of the two numbers.

How do you insert n arithmetic means between two numbers?

Find d = (b − a) ÷ (n + 1). Then the means are a + d, a + 2d, up to a + nd. The list from a to b forms an AP with n + 2 terms.

Why is it n + 1 and not n in the formula for d?

With n means plus the two end numbers there are n + 2 terms. The number of gaps between consecutive terms is one less than that, which is n + 1.

Is the sum of inserted means the same as the sum of the AP?

No. The sum of the inserted means leaves out a and b. It equals n(a + b) ÷ 2. The full AP sum adds a + b to that.