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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

1,3,5,7,9,11,etc......

 

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.

 

Example

If the general term of a sequence is given by 2n, Find the First, sixth and 8 th term.

 

Given Tn = 2n

Therefore

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

Also

 

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

= 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

 

Solved Examples

Question

Find the 10 th terms of the series:

(1)

(2)

(3)

(4)

 

Solution

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 =

[as ]

Hence = =

 

Question

 

If the 8 th term of an AP is 88 and the 88 th term is 8, then 100 th term is :

(1) 1

(2) 2

(3) -4

(4) 8

 

Solution

 

Given 8 th = 88 and 88 th term = 8, let the first term of AP be 'a' and common difference be 'd'.

then and

88 = a + 7d (1)

and 8 = a + 87d (2)

Subtracting we get 80 = -80d

d = 1

and putting (1) we get a = 95

Hence,

= a + 99d

= 95 + 99(-1)

= -4

Hence option (3)

 

Question

 

In the AP 3,7,11,15..... up to 50 terms and 2,5,8.....up to 50 terms, how many terms are identical ?

(1) 12

(2) 4

(3) 16

(4) 18

 

Solution

 

For the AP 3,7,11,15....

and

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 and

Hence and

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.

 

Question

 

The product of the first terms of the G.P having the third term as 24 as -

(1) 16

(2) 32

(3) 64

(4) 128

 

Solution

 

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

Hence (2)

Progressions, Sequences and Series ( Refresher test 2 ) Launch Test Discuss
Refresher Test - Progressions, Sequences and Series Launch Test Discuss

Practice Test - Progressions and Sequences Launch Test Discuss

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