Arithmetic Progression
An arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
For example, the sequence 3, 5, 7, 9, 11,... is an arithmetic progression with common difference 2.
Arithmetic progression property:
a1 + an = a2 + an-1 = ... = ak+an-k+1
Formulae for the n-th term can be defined as:
an = 1/2 x (an-1 + an+1)
If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the n-th term of the sequence is given by
an = a1 + (n - 1)d, n = 1, 2, ...
The sum S of the first n values of a finite sequence is given by the formula:
S = 1/2(a1 + an)n, where a1 is the first term and an the last.
or
S = 1/2(2a1 + d(n-1))n
Example 1: Find the sum of the first 10 numbers from this arithmetic progression 1, 11, 21, 31...
Solution: we can use this formula S = 1/2(2a1 + d(n-1))n
S = 1/2(2.1 + 10(10-1))10 = 5(2 + 90) = 5.92 = 460
Example 2: The sum of the three numbers in A.P is 21 and the product of their extremes is 45. Find the numbers.
Solution: Let the numbers are be a - d, a, a + d
Then a - d + a + a + d = 21
3a = 21
a = 7
and (a - d)(a + d) = 45
a2 - d2 = 45
d2 = 4
d = +2
Hence, the numbers are 5, 7 and 9 when d = 2 and 9, 7 and 5 when d = -2. In both the cases numbers are the same.
Geometric Progression
A geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence.
a geometric sequence can be written as:
Formulae for the n-th term can be defined as:
an = a1.qn-1
The common ratio then is:
q = |
|
A sequence with a common ratio of 2 and a scale factor of 1 is 1, 2, 4, 8, 16, 32...
A sequence with a common ratio of -1 and a scale factor of 3 is 5, -5, 5, -5, 5, -5,...
If the common ratio is:
- Negative, the results will alternate between positive and negative.
- Greater than 1, there will be exponential growth towards infinity (positive).
- Less than -1, there will be exponential growth towards infinity (positive and negative).
- Between 1 and -1, there will be exponential decay towards zero.
- Zero, the results will remain at zero
Geometric Progression Properties
a1.an = a2.an-1 =...= ak.an-k+1
Formula for the sum of the first n numbers of geometric progression
Sn = | a1 - anq 1 - q | = a1. | 1 - qn 1 - q |
Infinite geometric series where |q| < 1
If |q| < 1 then an -> 0, when n -> ∞ So the sum S of such a infinite geometric progression is:
S = |
|
Harmonic Progressions
Note:
i) The series formed by the reciprocals of the terms of a geometric series is also a geometric series.