When two numbers, 'a' and 'b', are given and 'a', 'a1', 'a2', 'a3', ..., 'an', 'b' form an arithmetic progression (AP), then 'a1', 'a2', 'a3', ..., 'an' represent 'n' arithmetic means (AMs) between 'a' and 'b'. To find these arithmetic means, we use the formula A1 = a + d, A2 = a + 2d, ..., An = a + nd, where 'd' is the common difference between consecutive terms, calculated as 'b - a(n + 1)'.
It is important to note that the sum of 'n' AMs inserted between 'a' and 'b' equals 'n' times a single AM between 'a' and 'b'. This property aids in finding the sum of AMs within an arithmetic progression and highlights the relationship between consecutive terms in the sequence.
Geometric Mean
Geometric Mean between Two Numbers
In a geometric progression (GP) where 'a', 'b', and 'c' are in GP, 'b' is termed the geometric mean (GM) between 'a' and 'c'. Mathematically, this relationship is represented by the equation
𝑏2=𝑎𝑐
b
2
=
a
c
, or alternatively,
𝑏=𝑎𝑐
b
=
a
c
, given that 'a' and 'c' are positive.
n-Geometric Means between Two Numbers
When 'a' and 'b' are two numbers, and 'a', 'G1', 'G2', 'G3', ..., 'Gn', 'b' form a geometric progression (GP), then 'G1', 'G2', 'G3', ..., 'Gn' represent 'n' geometric means (GMs) between 'a' and 'b'. These GMs are calculated using the formula
𝐺1=𝑎𝑟,𝐺2=𝑎𝑟2,...,𝐺𝑛=𝑎𝑟𝑛−1
G
1
=
a
r
,
G
2
=
a
r
2
,
...
,
G
n
=
a
r
n
−
1
, where 'r' is calculated as
(𝑏𝑎)1𝑛+1
(
a
b
)
n
+
1
1
.
An important property to note here is that the product of 'n' GMs inserted between 'a' and 'b' is equal to the nth power of a single GM between 'a' and 'b', denoted as
∏𝑟=1𝑛𝐺𝑟=𝐺𝑛
∏
r
=
1
n
G
r
=
G
n
.
Arithmetic, Geometric, and Harmonic Means between Two Given Numbers
Let 'A', 'G', and 'H' be the arithmetic, geometric, and harmonic mean between two integer numbers 'a' and 'b'. These means are calculated as
𝐴=𝑎+𝑏2
A
=
2
a
+
b
,
𝐺=𝑎𝑏
G
=
ab
, and
𝐻=2𝑎𝑏𝑎+𝑏
H
=
a
+
b
2
ab
respectively.
These means satisfy the properties
𝐴≥𝐺≥𝐻
A
≥
G
≥
H
,
𝐺2=𝐴𝐻
G
2
=
A
H
, indicating that they form a GP. Furthermore, the quadratic equation
𝑥2−2𝐴𝑥+𝐺2=0
x
2
−
2
A
x
+
G
2
=
0
has 'a' and 'b' as its roots.
Properties of Arithmetic & Geometric Means between Two Quantities
If 'A' and 'G' are arithmetic and geometric means between 'a' and 'b', then the quadratic equation
𝑥2−2𝐴𝑥+𝐺2=0
x
2
−
2
A
x
+
G
2
=
0
has 'a' and 'b' as its roots.
If 'A' and 'G' are AM and GM between two numbers 'a' and 'b', then
𝑎=𝐴+𝐴2−𝐺2
a
=
A
+
A
2
−
G
2
and
𝑏=𝐴−𝐴2−𝐺2
b
=
A
−
A
2
−
G
2
. These properties are crucial in understanding the relationships between different types of means and their applications in various mathematical contexts.
Sigma Notations
Theorems
(i)
∑
r
=
1
n
(
a
r
+
b
r
)
=
∑
r
=
1
n
a
r
+
∑
r
=
1
n
b
r
∑𝑟=1𝑛(𝑎𝑟+𝑏𝑟)=∑𝑟=1𝑛𝑎𝑟+∑𝑟=1𝑛𝑏𝑟
(ii)
∑
r
=
1
n
k
a
=
k
∑
r
=
1
n
a
r
∑𝑟=1𝑛𝑘𝑎=𝑘∑𝑟=1𝑛𝑎𝑟
(iii)
∑
r
=
1
n
k
=
n
k
∑𝑟=1𝑛𝑘=𝑛𝑘
Sum of n Terms of Some Special Sequences
Sum of first n natural numbers
∑
k
=
1
n
k
=
1
+
2
+
3
+
.
.
.
.
.
.
.
+
n
=
n
(
n
+
1
)
2
∑𝑘=1𝑛𝑘=1+2+3+.......+𝑛=𝑛(𝑛+1)2
Sum of squares of first n natural numbers
∑
k
=
1
n
k
2
=
1
2
+
2
2
+
3
2
+
.
.
.
.
.
.
.
+
n
2
=
n
(
n
+
1
)
(
2
n
+
1
)
6
∑𝑘=1𝑛𝑘2=12+22+32+.......+𝑛2=𝑛(𝑛+1)(2𝑛+1)6
Sum of cubes of first n natural numbers
∑
k
=
1
n
k
3
=
1
3
+
2
3
+
3
3
+
.
.
.
.
.
.
.
+
n
3
=
[
n
(
n
+
1
)
2
]
2
=
[
∑
k
=
1
n
k
]
2
Arithmetico-Geometric series
An arithmetic-geometric progression (A.G.P.) is a sequence where each term is the product of the terms of an arithmetic progression (AP) and a geometric progression (GP). This means that each term in the sequence can be expressed as the product of corresponding terms from an AP and a GP.
A
P
:
1
,
3
,
5
,
.
.
.
.
.
.
.
.
.
.
𝐴𝑃:1,3,5,..........
and
G
P
;
1
,
x
,
x
2
,
.
.
.
.
.
.
.
.
𝐺𝑃;1,𝑥,𝑥2,........
A
G
P
:
1
,
3
x
,
5
x
2
,
.
.
.
.
.
.
.
.
⇒𝐴𝐺𝑃:1,3𝑥,5𝑥2,........
Sum of n terms of an Arithmetico-Geometric Series
S
n
=
a
+
(
a
+
d
)
r
+
(
a
+
2
d
)
r
2
+
…
…
+
S𝑛=a+(a+d)r+(a+2d)r2+……+
[
a
+
(
n
−
1
)
d
]
r
n
−
1
[𝑎+(𝑛−1)𝑑]𝑟𝑛−1
S
n
=
a
1
−
r
+
d
r
(
1
−
r
n
−
1
)
(
1
−
r
)
2
−
[
a
+
(
n
−
1
)
d
]
r
n
1
−
r
,
r
≠
1
𝑆𝑛=𝑎1−𝑟+𝑑𝑟(1−𝑟𝑛−1)(1−𝑟)2−[𝑎+(𝑛−1)𝑑]𝑟𝑛1−𝑟,𝑟≠1
Sum to Infinity
If
|
r
|
<
1
\&
n
→
∞
|𝑟|<1 \& 𝑛→∞
, then
lim
n
→
∞
=
0.
S
∞
=
a
1
−
r
+
d
r
(
1
−
r
)
2
lim𝑛→∞=0.𝑆∞=𝑎1−𝑟+𝑑𝑟(1−𝑟)2
.
