Arithmetic Progressions
Quantities are said to be in arithmetic progression when they increase or decrease by a common difference. Thus each of the following series form an arithmetic progression:
3,7,11,15.. a, a+d, a+2d, a+3d…
The common difference is found by subtracting any term of the series from the next term.
To find the Sum of the given number of Terms in a Arithmetic Progression
S = (n(a+L))/2
L= a + (n-1)d
S= n/2 * (2a+(n-1)d)
Let a denote the first term, d the common difference, and n the total number of terms. Also L is denoted as the last term and S the required sum.
When three quantities are in arithmetic progression, the middle one is said to be the arithmetic mean of the other two. Thus a is the arithmetic mean between a – d and a + d.
To Find the Arithmetic Mean between any two given quantities
Let a and b be two quantities and A be their arithmetic mean. Then since a, A, b are in A.P. we must have b-A = A-a. Each being equal to the common difference: This gives us
A=(a+b)/2
Two insert a given number of Arithmetic Means between Two given Quantities:
Let a and b be two given quantities and n be the number mean. Let d be the common difference. Hence d = (b-a)/(n+1).
And the required means are a+(b-a)/(n-1), a+2(b-a)/(n+1), … a+n(b-1)/n+1
Geometric Progression
Quantities are said to be in geometric progression when they increase of decrease by a constant factor. If we examine a series a, ar, arˆ2, ar^3,…. Let n be the number of terms and if l denote the last, or the nth term, we have
l = ar^(n-1).
When three quantities are in geometric progression, the middle one is called the geometric mean between the other two.
To find the Geometric Mean between two given quantities
Let a and b be two quantities, G the geometric mean. Then since a, G, b are in G.P. Hence G = √ab.
To find the Sum of a Number of Terms in a Geometric Progression
Let a be the first term, r the common ratio, n the number of terms and Sⁿ be the sum of n terms.
if r>1,
Sⁿ = (a(rⁿ-1)) / (r-1)ⁿ,
if r<1,
then Sⁿ = (a(rⁿ-1))/(r-1)
Sum of an infinite geometric progression where r < 1
S∞ = a/(1-r)
This formulae is used when common ratio of G.P is than one.
Harmonic Progression
Three quantities a, b, c are said to be in Harmonic Progression when a/c =(a-b)/(b-c)
To find the Harmonic Mean between two given quantities
Let a, b be the two quantities, H their harmonic mean; then 1/a, 1/H, 1/b are in A.P
H = 2ab/(a+b)