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CHAPTER
12
An arithmetic progression (A
...
) : a, a + d, a + 2d
...
P
...
P
...
P
...
P
...
Properties of A
...
(i) If a, b, c are in A
...
P
...
(ii) Three numbers in A
...
can be taken as a – d, a, a + d; four
numbers in A
...
can be taken as a – 3d, a – d, a + d, a + 3d;
five numbers in A
...
are a – 2d, a – d, a, a + d, a + 2d & six
terms in A
...
are a – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d etc
...
P
...
Arithmetic Mean (Mean or Average) (A
...
):
If three terms are in A
...
then the middle term is called the A M
...
P
...
M
...
n–Arithmetic Means Between Two Numbers:
If a, b are any two given numbers & a, A1 A2
...
P
...
An are the A
...
2 (b − a )
n (b − a )
b−a
A1 =
a+
a+
a+
, A2 =
, …
...
between a & b
...
with a as the first
term & r as common ratio
...
e
...
):
If a
...
b is the G M between a & c, then b2 = ac
n Geometric Means Between Positive Number a, b: If a, b are
two given numbers & a
...
,G3 b are in G
...
, Gn are n GM
...
G2 = a (b/a)2/n+1
...
M
...
, b is the H
...
between a & c, then b =
2ac
a+c
1 11 1
1
H
...
an is given by=
+ +……
...
M
...
M
...
M
...
M
...
M
...
M
...
= an
Important Results
n
n
n
(i)
∑ ( ar ± br ) =
∑ ar ± ∑ br
=r 1 =
r 1
=
r 1
n
(ii)
∑ kar = k
n
∑ ar
=r 1 =r 1
n
(iii)
∑ k = nk where k is a constant,
r =1
n
∑ r=
(iv)
r =1
(v)
n
∑ r=
2
1 + 2 + 3 +………
...
+ n=
r =1
(vi) ∑ r 3= 13 + 23 + 33 +………
...
+ an ) − ( a1 2 +a2 2 +……+ an2 )
1
i< j =