Arithmetic Mean (A.M.)

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1. If a, x, b are in A.P, then x is called the arithmetic mean (A.M.) between the extremes a and b.

2. a. To insert n arithmetic means between two given quantities.

Let a and b be any two given quantities, and let A₁,A₂,A₃,-----A_n be n arithmetic means to be inserted between a and b so that a, A₁,A₂,A₃,-----A_n, b will be in A.P.

Let d be the common difference.

Now the total number of terms including a and b is (n+2)

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Hence the n arithmetic means are:

These can be inserted between a and b.

b. To find the sum of n arithmetic means between the two given quantities:

Let the two quantities (numbers) be a and b.

The required sum = A₁ + A₂ + A₃ + ---- + A_n