Progressions
Progressions
A set of numbers in which one number is connected to the next number by some law is called a series or a progression.
Arithmetic progressions
The relationship between consecutive numbers in an arithmetic progression is that they are connected by a common difference. For the set of numbers 3, 6, 9, 12, 15, …, the series is obtained by adding 3 to the preceding number; that is, the common difference is 3. In general, when a is the first term and d is the common difference, the arithmetic progression is of the form:
Term Value
1st a
2nd a+d
3rd a+2d
4th a+3d,…
last a+(n-1)d
where:
n is the number of terms in the progression.
The sum Sn of all the terms is given by the average value of the terms times the number of term; that is:
Sn=[(first+last)/2] ×(number of terms)
=[(a+a+(n-1)d/2] ×n
=(n/2)[2a+(n-1)d]
Geometric progressions
The relationship between consecutive numbers in a geometric progression is that they are connected by a common ratio. For the set of numbers 3, 6, 12, 24, 48, …, the series is obtained by multiplying the preceding number by 2. In general, when a is the first term and r is the common ratio, the geometric progression is of the form:
Term 1st 2nd 3rd 4th last
Value a ar ar2.. ar3,…. arn-1
where:
n is the number of terms in the progression.
The sum Sn of all the terms may be found as follows:
Sn=a+ar+ar2+ar3+…..+arn-1 (1)
Multiplying each term of equation (1) by r gives
rSn=ar+ar2+ar3+….+arn-1+arn (2)
Subtracting equations (2) from (1) gives:
Sn(1-r)=a-arn (3)
Sn=[a(1-rn)]/(1-r)
Alternatively, multiplying both numerator and denominator by -1 gives:
Sn=[a(rn-1)]/(r-1) (4)
It is usual to use equation (3) when r <1 and (4) when r >1.
When -1 > r >1, each term of a geometric progression is smaller than the preceding term and the terms are said to converge. It is possible to find the sum of all the terms of a converging series. In this case, such a sum is called the sum to infinity. The term [a(1-r n)]/(1-r) can be rewritten as [a/(1-r)]- [ar n/(1- r)]. Since r is less than 1, r n becomes smaller and smaller as n grows larger and larger. When n is very large, rn effectively becomes zero, and thus [ar n/(1 –r)] becomes zero. It follows that the sum to infinity of a geometric progression is a/(1 –r), which is valid when -1 >r >1.
Harmonic Progressions
The relationship between numbers in a harmonic progression is that the reciprocals of consecutive terms form an arithmetic progression. Thus for the arithmetic progression 1, 2, 3, 4, 5, …, the corresponding harmonic progression is 1, 1/2, 1/3, 1/4, 1/5, ….