Free Online Preparation for CAT with Minglebox e-CAT Prep. Cover basic concepts of Progressions, Sequences and Series under Quantitative Aptitude for MBA Entrance Exam Preparation with Study material, solved examples and tests prepared by CAT coaching experts.
A progression or sequence is defined as a succession of terms arranged in a definite according to some rule.
For example the sequence of Odd numbers
The numbers in the sequence are called the terms of the sequence. A sequence having a finite number of terms is known as finite sequence.
The first terms in a sequence is generally denoted by a, second term by or
. The term of the sequence is denoted by or and is also known as
the general term of the sequence, as by assigning value to 'n' , can be made to represent any of the terms in the sequence.
If the general term of a sequence is given by 2n, Find the First, sixth and 8 th term.
Given Tn = 2n
First Term n=1
Therefore =2.1 = 2
Similarly = 2.6 = 12 and
= 2.8 = 16
Some Standard type of sequences:
Arithematic Progression (AP)
In this type of sequence, the difference between any two consecutive terms of the sequence is a constant. The constant is known as the 'Common difference.' Thus if the first term in AP is a, and Common difference is 'd' then second term is given by = a+d, third terms = a+2d etc.....
in general = a+(n-1)d.
Sum of n terms of an AP is given by
Geometric Progression (GP)
In this type of sequence the ratio of 2 consecutive terms is a constant. The constant ratio is know as common ration and is usually denoted by 'r'. Thus if 'a' is the first terms of a GP
then = ar , = etc.
Sum of n terms of a GP, if
Sum up to infinite terms of a GP
If then ------->0 as n-----> i.e. if lets say then if we consider higher and
higher values of n then becomes smaller and smaller. Hence if we consider the sum up to very
large number of terms, we say the sum up to infinite terms,
Harmonic Progression (HP)
A sequence is said to be a Harmonic Progression
if are in AP.
The term of HP is given by
There is no formula for finding the sum up to n terms of a HP.
Arithmetic mean (AM) of 2 positive numbers a,b is defined as
Geometric mean (GM) of 2 positive numbers a,b is defined as
Harmonic mean (HM) of 2 positive numbers a,b is defined as . For any given a,b
Find the 10 th terms of the series:
Term of the series can be split as
We find 4,3,2....... are in AP with common difference of -1 hence 10 th term is
given by = 4 + (10-1)(-1) = -5
[as = a +(n – 1) d ]
Also are in GP
Hence = =
If the 8 th term of an AP is 88 and the 88 th term is 8, then 100 th term is :
Given 8 th = 88 and 88 th term = 8, let the first term of AP be 'a' and common difference be 'd'.
88 = a + 7d (1)
and 8 = a + 87d (2)
Subtracting we get 80 = -80d
d = 1
and putting (1) we get a = 95
= a + 99d
= 95 + 99(-1)
Hence option (3)
In the AP 3,7,11,15..... up to 50 terms and 2,5,8.....up to 50 terms, how many terms are identical ?
For the AP 3,7,11,15....
and for the AP 2,5,8,.......
if terms of 1 st AP is equal to the term of Second AP, then
4n-1 = 3m-1
4n = 3m or 4n = 3m
= x Say.
As x has to be an integer
For x = 1,2,3......12
Thus there are 12 terms in the 2 sequences which are equal.
The product of the first terms of the G.P having the third term as 24 as -
Let the third term be 'a' then first, second fourth and fifth term will be given as
where r is the common ratio.
Now product of first five terms =
but a = 2
product = =32