Master Series and Progressions Aptitude with These Tricks and Formulas for Competitive Exams

Master Series and Progressions Aptitude with These 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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