Definition
A sequence
(progression) is a set of numbers in a definite order with a definite rule of
obtaining the numbers.
Arithmetic Progression (A.
P.)
Definition
An
A.P. is a sequence whose terms increase or decrease by a fixed number, called
the common difference of the A.P. .
Term and Sum of n Terms:
If a is the first
term and d the common difference, the A.P. can be written
as a, a+d, a + 2d, ...... . The nth term an is given by an
= a + (n - 1)d. The sum Sn of the first n terms of such an A.P. is
given by 
(a + l ) where l is
the last term (i.e. the nth term of the A.P.).
Notes:
- If a fixed number is
added (subtracted) to each term of a given A.P. then the resulting sequence is
also an A.P. with the same common difference as that of the given A.P.
- If each term of an
A.P. is multiplied by a fixed number(say k) (or divided by a non-zero fixed
number), the resulting sequence is also an A.P. with the common difference
multiplied by k.
- If a1, a2,
a3.....and b1, b2, b3...are two
A.P.’s with common differences d and d¢
respectively then a1+b1, a2+b2, a3+b3,...is
also an A.P. with common difference d+d¢
.
- If we have to take
three terms in an A.P., it is convenient to take them as a - d, a,
a + d. In general, we take a - rd, a - (r - 1)d,......a - d, a, a +
d,.......a + rd in case we have to take (2r + 1) terms in an A.P. - If we have to take
four terms, we take a - 3d, a - d, a + d, a + 3d. In general, we take a - (2r -
1)d, a - (2r - 3)d,....a - d, a + d,.....a + (2r - 1)d, in case we have to
take 2r terms in an A.P.
- If a1, a2,
a3, ……. an are in A.P. then a1 + an
= a2 + an-1 = a3 + an –2 = . . . .
. and so on.
Arithmetic Mean(s):
- If three terms are
in A.P., then the middle term is called the arithmetic mean (A.M.) between the
other two i.e. if a, b, c are in A.P. then
is the A.M. of a
and c. - If a1, a2,
... an are n numbers then the arithmetic mean (A) of these numbers
is
- The n numbers A1, A2......An
are said to be A.M.’s between the numbers a and b if.
- a, A1, A2,........An,b
are in A.P. If d is the common difference of this A.P. then.
- b = a + (n + 2 - 1)d
&⇒
&⇒ 
where Ar
is the rth mean.
Example 1
If the Ist
and the 2nd terms of an A.P are 1 and –3 respectively, find the nth
term and the sum of the Ist n terms.
Solution:
Ist
term = a, 2nd term = a + d where a = 1, a + d = -3,
&⇒ d = – 4 (Common difference of A.P.).
we have an
= a + (n –1)d
= 1 + (n – 1) (– 4)
= 5 – 4n
Sn = 
{a + an}
= {1 + 5 – 4n} = n (3
– 2n)
Example 2
If 6 arithmetic means
are inserted between 1 and 9/2, find the 4th arithmetic mean.
Solution:
Let a1,
a2, a3, a4, a5, a6 be
six arithmatic means .
Then 1, a1,
a2, …, a6, 
will be in A.P.
Now, = 1 + 7d
&⇒ 
= 7d
&⇒ d = 
Hence a4
= 1 + 4
= 3
