Series and Progressions Aptitude Tricks and Formulas for
Competitive Exams

Hello friends, today we are sharing a series and progressions aptitude
tricks and formulas article for various competitive exams that can be used to
give a good performance in the upcoming exams.

Arithmetic
Progression: 

An Arithmetic
Progression (AP) or arithmetic sequence is a sequence of numbers such
that the difference between the consecutive terms is constant. 

For instance, the sequence 5, 7, 9, 11, 13, 15 … is an arithmetic progression
with common difference of 2.

Its general form can be given as a, a+d, a+2d, a+3d,…

If the initial term of an arithmetic progression is a and the
common difference of successive members is d, then the 
nth term of the sequence (a_n) is given by: 

a_n =
a + (n – 1)d

and in
general 

Nth Term
of A.P. is A_n = a_m + (n – m)d

The sum
of the members of a finite arithmetic progression is called an arithmetic
series and given by, 

Sum of N
terms of an A.P. is S_n = ⁿ/₂ [2a
+ (n – 1)d]  = ⁿ/₂ (a + l)

Arithmetic
mean: 

When three
quantities are in AP, the middle one is said to be the Arithmetic Mean (AM) of
the other two, thus a is the AM of (a-d) and (a+d).

Arithmetic
mean between two numbers a and b is given by, 

AM =
 (a+b)/₂

 

Geometric
progression:

A geometric
progression, also known as a geometric sequence, is a sequence of numbers where
each term after the first is found by multiplying the previous one by a fixed,
non-zero number called the common ratio.

For example,
the sequence 2, 6, 18, 54, … is a geometric progression with common ratio
3. 

 

The general
form of a geometric sequence is a, ar, ar²,ar³,ar⁴,…

 

A geometric
series is the sum of the numbers in a geometric progression.

 

Let a be
the first term and r be the common ratio, a_n nth
term, n the number of terms, and S_n be the
sum up to n terms:

 

The n-th
term  is given by, 
a_n = ar^n-1

The Sum
up to n-th term of Geometric progression (G.P.)  is
given by,

If r >
1, then

S_n =
a(rⁿ-1)/(r-1)

 

if r <
1, then

S_n =
a(1-rⁿ)/(1-r)

 

Sum of
infinite geometric progression when r<1: 

S_n =
a/(1-r)

 

Geometric
Mean (GM) between two numbers a and b is given by, 

GM = sqrt
ab 

 

Some
useful results on number series:

Sum of
first n natural numbers is given by

S = 1 + 2 +
3 + 4 +….+n 

S = n/2 *
(n+1) 

 

Sum of
squares of the first n natural numbers is given by

S = 1² +
2² + 3² +….+n²

S =
[{n(n+1)(2n+1)}/6 ]

 

Sum of
cubes of the first n natural numbers is given by

S = 1³ +
2³ + 3³ +….+n³

S =
[{n(n+1)}/2]

 

Sum of
first n odd natural numbers 

S = 1 + 3 +
5 +…+ (2n-1)

S
= n²

 

Sum
of first n even natural numbers S = 2 + 4 + 6 +…+ 2n

S =
n(n+1)

Note: 

1) If we are
counting from n1 to n2 including both the end points, we get (n2-n1) +
1 numbers.

e.g. between
12 and 22, there is (22-12) +1 = 11 numbers (Including both the ends).

2) In the
first n, natural numbers:

i) If n
is even

There
are n/2 odd and n/2 even numbers 

e.g from 1
to 40 there are 25 odd numbers and 25 even numbers.

ii) If n is
odd

There
are (n+1)/2 odd numbers, and (n-1)/2 even
numbers

e.g. from 1
to 41, there are (41+1)/2= 21 odd numbers and (41-1)/2 = 20 even numbers.

 

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