Progressions
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PROGRESSIONS
ARITHMETIC PROGRESSION
A sequence of numbers is in arithmetic progression (AP) when the difference between two consecutive numbers is a constant. We can thus say that tn+1 - tn = d. This constant d is called the common difference of the arithmetic progression. In other words, an arithmetic progression can be described as a series of numbers, which increase or decrease, by the same constant.
The nth term of an arithmetic progression is denoted by tn and is given by
| tn = a + (n-1)d |
NOTE: if each term of an AP is increased, decreased, multiplied or divided by the same number (real and non-zero), then the resulting series is also in AP. In the case of addition and subtraction, the resulting series has the same common difference.
NOTE: if a1, a2, a3,….. and b1, b2, b3,…… be two arithmetic progressions then the progression a1+ b1, a2+ b2, a3+ b3 + ……., is also an arithmetic progression.
SUM OF AN ARITHMETIC PROGRESSION
The sum of an arithmetic progression till the nth term is given by
| Sn = ( a+l ) n/2 |
| Sn = n/2[ 2a+( n-1 )d ] |
ARITHMETIC MEAN
If two numbers a and b be in AP then the arithmetic mean between them is given as
| A = (a+b)/2 |
NOTE: the numbers a, A and b are in AP
If a1, a2, a3, a4,………, an are n numbers in AP then their arithmetic mean is given by
| A = (a1+ a2+ a3+ a4+………+ an )/n |
n- ARITHMETIC MEANS
The numbers A1, A2, A3, A4,………, An are said to be arithmetic means between the numbers a and b if the series
a, A1, A2, A3, A4,………, An ,b
is an AP. The nth mean An is given as
| An = a + n( b - a )/ (n + 1) |
GEOMETRIC PROGRESSION
A series is in geometric progression if each of its succeeding number is r times its preceding term (the first term being non-zero). The number r is called the common ratio and it is a constant, non-zero number. Thus, we can define a GP as a series in which each term is same multiple of the preceding term.
The nth term of a geometric progression is denoted by tn and is given by
| tn = arn-1 |
NOTE: if each term of a GP is multiplied or divided by the same number (real and non-zero), then the resulting series is also in GP.
SUM OF A GEOMETRIC PROGRESSION
The sum of a geometric progression till the nth term, if r > 1, is given by
|
Sn = a(rn - l )/ (r-1) |
The sum of a geometric progression till the nth term, if r < 1, is given by
| Sn = a(1- rn )/ (1- r) |
| Sn = n/2[ 2a+( n-1 )d ] |
GEOMETRIC MEAN
If two numbers a and b be in GP then the geometric mean between them is given as
| G = (ab)1/2 |
NOTE: if a1, a2, a3,……..and b1, b2, b3,…… be two geometric progressions then the progression a1b1, a2b2, a3b3 ,……., is also an geometric progression.
n- GEOMETRIC MEANS
The numbers G1, G2, G3, G4,………, Gn are said to be geometric means between the numbers a and b if the series
a, G1, G2, G3, G4,………, Gn ,b
is a GP. The nth mean Gn is given as
| Gn = arn |
SUM OF AN INFINITE GEOMETRIC PROGRESSION
If a geometric progression be given as a + ar + ar2+ ar3 ……..till infinity, then the sum of such a series is given as
| S = a/(1-r) |