# Coding And Number Series Formulas

## Formulas For Coding And Number Series in Logical

Here , in this Page Formulas For Coding and Number Series is given. We have also covered its types along with questions and answers. Coding questions are very easy to solve once we get to know about the tricks behind them. ## Coding and Number Series Formulas

Arithmetic ProgressionGeometric Progression
Sequencea, a+d, a+2d,……,a+(n-1)d,….a, ar, $ar^{2}$,….,$ar^{(n-1)}$,…
Common Difference or Ratio

Successive term – Preceding term

Common difference = d = a2 – a1

Successive term/Preceding term
Common ratio = $r = ar^{(n-1)}/ar^{(n-2)}$
General Term (nth Term)an = a + (n-1)dan = $ar^{(n-1)}$
nth term from the last terman = l – (n-1)dan = $l/r^{(n-1)}$
Sum of first n termssn = n/2(2a + (n-1)d)sn = a(1 – rn)/(1 – r) if |r| < 1
sn = a(rn -1)/(r – 1) if |r| > 1

### Types of Coding and Number Series

• Arithmetic Sequences and Series
• Geometric Sequences and Series
• Harmonic Sequences and Series
• Fibonacci Numbers

An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is always the same. This constant difference is called the “common difference.” In other words, an arithmetic series is a sum of the terms in an arithmetic progression.

For example, consider the arithmetic progression: 2, 5, 8, 11, 14, …

Here, the common difference is 3 (5 – 2 = 3, 8 – 5 = 3, and so on).

The formula to find the sum of the first n terms of an arithmetic series is given by:

S_n = n/2 * [2a + (n – 1)d]

A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the “common ratio.” In other words, a geometric series is a sum of the terms in a geometric progression.

For example, consider the geometric progression: 2, 6, 18, 54, …

Here, the common ratio is 3 (6 / 2 = 3, 18 / 6 = 3, and so on).

The formula to find the sum of the first n terms of a geometric series is given by:

S_n = a * (1 – r^n) / (1 – r)

The harmonic series is an infinite series that is formed by adding the reciprocals of the natural numbers. It has the following form:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + …

In other words, the nth term of the harmonic series is given by 1/n. The harmonic series diverges, which means that it does not have a finite sum. As you add more and more terms to the series, the sum keeps getting larger and larger without bound.

The Fibonacci series is a sequence of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. So, the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

Mathematically, the Fibonacci series is defined by the recurrence relation:

• F(n) = F(n-1) + F(n-2)

Where F(n) is the nth Fibonacci number, F(n-1) is the (n-1)th Fibonacci number, and F(n-2) is the (n-2)th Fibonacci number.

The first few terms of the Fibonacci series are:

• F(0) = 0
• F(1) = 1
• F(2) = 1
• F(3) = 2
• F(4) = 3
• F(5) = 5

### Coding and Number Series Questions and Answer with Formulas

Question 1:
In a certain code ‘NOTE’ is written as ‘TNEO’ and ‘SALE’ is written as ‘LSEA’. How will ‘SEAT’ be written in this code?

Explanation:

Here, NOTE = TNEO
SALE = LESA
Analyze the pattern and the set of reasoning trick
SALE => LSEA
On a close examination, we notice;

N-1 O-2 T-3 E-4 => T-3 E-1 N-4 O-2

S-1 A-2 L- 3 E-4 => L-3 S-1 E-4 A-2

So applying the same order to the given word we get-
SEAT as ASTE

Question 2:
In a certain code ‘5892’ is written as ‘2598’ and ‘2649’ is written as ‘9246’. How will ‘7293’ be written in this code?

Explanations

5892 -> 2598

2649 -> 9246

After observing we see 1st is at Second Position , 2nd is at Last Position , 3rd Position remains same , last is at First Position

7293 -> 3792

Question 3:
In a certain code ‘365’ is written as ‘698’. How will ‘264’ be written in this code?

Explanation:

365 – > 698

After Observing we see Each digit increases by 3

so 264 ->  597

Question 4 :
In a certain code language, if the word ‘GUARDIAN’ is coded as ‘AAIUGRDN’ and ‘ FANTASTIC’ Is coded as ‘ AAIFNTSTC’ Then what is the code for the word ‘GALAXY’ ?

Explanation:

In word GUARDIAN the vowels are taken first in the same order as they are in the word followed by the consonants So the code becomes AAIUGRDN
Similarly, In word GALAXY the vowels are taken first in the same order as they are in the word followed by the consonants So the code becomes AAGLXY.

Question 5:
If ARYAN is written as XOVXK , then what is the code of NIKHIL ?

Explanation :

Each alphabet of the word is moved 3 steps backward.

In the same way NIKHIL will be written as KFHEFI

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