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Arithmetic Mean

The sum of all of the numbers in a list divided by the number of terms in that list gives
the arithmetic mean of that list. Let us first understand the concept of arithmetic
progression.

In the arithmetic progression, we know that if the three numbers are in AP, that means
if a, b and c are in AP, then basically the first two terms a and b will have the difference
which will be equal to the next two terms b and c.

So we can say, b – a = c – b. Rearranging the terms,
2b = a + c

b =

So we can say that this term b is the average of the other two terms a and c. This
average in arithmetic progression is called the arithmetic mean.

Arithmetic mean = A =

where A = arithmetic mean, N = the number of terms and S = the sum of the numbers
in the list.

Properties of Arithmetic Progression

(i) If a constant quantity is added to or subtracted from each term of an Arithmetic
Progression (A. P.), then the resulting terms of the sequence are also in A. P.
with same common difference (C.D.).
(ii) If each term of an Arithmetic Progression is multiplied or divided by a non-zero
constant quantity, then the resulting sequence form an Arithmetic
Progression.
(iii) In an Arithmetic Progression of finite number of terms the sum of any two terms
equidistant from the beginning and the end is equal to the sum of the first and
last terms.
(iv) Three numbers x, y, and z are in Arithmetic Progression if and only if 2y = x +
z.

Example 4- For the AP -3, – 7, – 11,… can we find directly a3 0 – a 20 without
actually finding a 30 and a 20? Give reason for your answer.

Solution-

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