An arithmetic progression is a sequence in which each term (except the first term) is obtained from the previous term by adding a constant known as the common difference. An arithmetic series is formed by the addition of the terms in an arithmetic progression.
Let the first term on an A. P. be a and common difference d. Then,
General form of an A. P.:
a, a + d, a + 2d, a + 3d, ...
nth term of an A. P.:
a + (n - 1) d
Sum of first n terms of an A. P.:
n/2 [2a + (n - 1) d]or
n/2 [ first term + last term]
This idea was from the mathematician Carl Friedrich Gauss, who, as a young boy, stunned his teacher by adding up 1 + 2 + 3 + ... + 99 + 100 within a few minutes. Here's how he did it:
He counted 101 terms in the series, of which 50 is the middle term. He also realised that adding the first and last numbers, 1 and 100, gives, 101; and adding the second and second last numbers, 2 and 99, gives 101, as well as 3 + 98 = 101 and so on. Thus he concluded that there are 50 sets of 101 and the middle term is 50. So the sum of the series is:
50 (1 + 100) + 50 = 5050.
This can be rewritten as:
100/2 (1 + 100) + 50 = 5050 or
101/2 (1 + 100) = 5050
Arithmetic mean. Given x, y and z are consecutive terms of an A. P., then
y - x = z - y
2y = x + z
y is known as the arithmetic mean.
Properties of A. P. (summary of the above points mentioned)
Given a sequence u1, u2, u3, ... un-1, un, un+1, ...
1. un is in the form a + (n - 1)d.
2. un - un-1 is a constant (common difference).
3. un+1 - un = un - un-1
Geometric Progressions
A geometric progression is a sequence in which each term (except the first term) is derived from the preceding term by the multiplication of a non-zero constant, which is the common ratio. A geometric series is formed by the addition of the terms in a geometric progression.
Examples:
1) 3, 6, 9, 12, ... first term 3, common ratio 3
2) 4, -8, 16, -32, ... first term 4, common ratio -2
Let the first term be a and common ratio be r.
General form of a G. P.:
a, ar, ar2, ar3, ...
nth term of a G. P.=
arn-1
Sum to first n terms of a G. P.:
Geometric mean. When x, y and z are consecutive numbers in a G. P.,
y2 = xz
y is the geometric mean
Properties of a G. P.(summary of the abovementioned points)
1. nth term is in the form arn-1, where a and r are constant
2. is constant for all n (common ratio).
3.
Sum to Infinity
The sum to infinity is a finite value the sum of the first n terms of a geometric series tends to when n tends to infinite. Sum to infinity only exists when a series is convergent.
Sum to infinity is given by the expression:
and only exists if :
Why?
The sum of first n terms is given by:
As . (only if )
Hence .
Thus the sum to infinity is given by
If , , and the sum to infinity will not exist.
Harmonic Progressions
A Harmonic Progression (HP) is is a series of terms where the reciprocals of the terms are in Arithmetic Progression (AP).
The general form of an HP is 1/a, 1/(a+d), 1/(a+2d), 1/(a+3d), .....
The nth term of a Harmonic Progression is given by tn=1/(nth term of the corresponding AP)
In the following Harmonic Progression
The Harmonic Mean (HM) of two numbers a and b is
The Harmonic Mean of n non-zero numbers is
Relation between AM, GM
& HM
that is, AM, GM, HM are in Geometric Progression.
For two positive numbers, AM ? GM ? HM equality holding for equal numbers.
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General form of an A. P.:
is constant for all n (common ratio).
. (only if 
.
, 
, and the sum to infinity will not exist.
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