# Series and Progressions Aptitude Tricks and Formulas for Competitive Exams

## 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:__

__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

*n*th 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, ar ^{2},ar^{3},ar^{4},…*

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**be the sum up to n terms:

_{n}**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(r^{n}-1)/(r-1)**

*if r <
1, then*

**S _{n} =
a(1-r^{n})/(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 = 1^{2} +
2^{2} + 3^{2} +....+n^{2}

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

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

S = 1^{3} +
2^{3} + 3^{3} +....+n^{3}

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

**Sum of
first n odd natural numbers **

S = 1 + 3 + 5 +...+ (2n-1)

**S
= n ^{2}**

**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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