These notes will be useful in solving some of the CAT problems that are related to progressions.
ARITHMETIC PROGRESSION
nth term of an Arithmetic progression = a + (n-1)d
Sum of n terms in an AP = s = n/2 [2a + (n-1)d]
where, a is the first term and d is the common differnce.
If a, b and c are any three consecutive terms in an AP, then 2b = a + c
GEOMETRIC PROGRESSION
nth term of a GP is = a[r^(n-1)]
sum of n terms of a GP:
s = a [(r^n - 1)/(r-1)] if r > 1
s = a [(1 - r^n)/(r-1)] if r < 1]
sum of an infinite number of terms of a GP is
s(approx.) = a/ (1-r) if r <1
If a, b and c are any three consequtive terms in a GP, then b^2 = ac
HARMONIC PROGRESSION
A series of non-zero numbers is said to be harmonic progression (abbreviated H.P.) if the series obtained by taking reciprocals of the corresponding terms of the given series is an arithmetic progression.
For example, the series 1 +1/4 +1/7 +1/10 +..... is an H.P. since the series obtained by taking reciprocals of its corresponding terms i.e. 1 +4 +7 +10 +... is an A.P.
A general H.P. is 1/a + 1/(a + d) + 1(a + 2d) + ...
nth term of an H.P. = 1/[a +(n -1)d]
Three numbers a, b, c are in H.P. only if 1/a, 1/b, 1/c are in A.P.
i.e. only if 1/a + 1/c = 2/b
i.e. only if b= 2ac/(a + c)
Thus the H.M. between a and b is H = 2ac/(a + c)
TIPS FOR SOLVING GMAT PROBLEMS:
If A, G, H are arithmetic, geometric and harmonic means between two distinct, positive real numbers a and b, THEN
1. G² = AH i.e. A, G, H are in G.P.
2. A, G, H are in descending order of magnitude i.e. A > G > H.