When three numbers and
are in A.P., then
is average of
and
.
Then
In general if are in A.P, their Arithmetic Mean A.M is defined as
Note -:
The terms are called the Arithmetic Means between
and
.
Some Facts about an A.P
(i) If a fixed number is added to (subtracted from) each term of a given AP, then the resulting sequence is also an AP and it has the same common difference as that of the given AP.
(ii) If each term of an AP is multiplied by a fixed constant (or divided by a non-zero fixed constant), then the resulting sequence is also an AP.
(iii) If and
are two Arithmetic Progressions, then
is also an AP.
(iv) If we have to take three terms in AP whose sum is known, it is convenient to take them as . In general, if we take
terms in AP, we take them as
.
(v) The numbers
are said to be Arithmetic Means between
and
, if
is an AP.
Here is the first term and
is the
term of the AP.
If is the common difference of this AP, then
Thus,