# What is the sum of first 30 natural numbers?

A progression is basically a list of terms ( usually numbers) that follow a particular logical and predictable pattern. There is a certain relation between the two terms in each type of Progression. The predictable nature of Progression helps in forming a generalized formula for that Progression, Formulae include finding the nth term of the series, finding the sum of the series, etc. There are three main types of progressions known,

**Types of Progression**

In Mathematics, the progression of numbers can be classified into three specific types mainly:

- Arithmetic Progression
- Geometric Progression
- Harmonic Progression

Let’s learn in detail about the arithmetic progression,

**Arithmetic Progression**

Arithmetic Progression is basically a sequence of numbers which exist in such a way that the difference between any two consecutive numbers is a constant value or quantity, that difference is denoted as “d”. The first term in A.P. is denoted as “a” and the last term (for finite series) as “n”. For instance, consider the sequence of even natural numbers 2, 4, 6, 8, 10,…….If we consider the difference between any two numbers (8- 6) is 2. Some of the other few examples of Arithmetic Progression are Sequence of odd natural numbers, Sequence of natural numbers.

**A generalized representation of Arithmetic Progression**

The first term is represented as “a” and the common difference is represented as “d”, therefore, the next term should be a+d, and the next term to that should be a+d+d, based on this, a generalized way of representing the A.P. can be formed. The Arithmetic Progression can be expressed as,

a, a+d, a+2d, a+3d, a+4d, ………. a+(n-1)d

In the above expression, “a” represents the first term of the progression, “d” represents the common difference

The last term “a_{n}” of the progression is represented as,

a

_{n }= a + (n-1)d

**What is the formula for the sum of n terms of an A.P?**

The Sum of any progression is basically the summation of all its terms, there is a generalized formula formed for the n terms of an A.P. If the first term is denoted as “a”, the common difference is denoted as “d”, the number of terms present is denoted as “n”, then the formula is given as,

S_n= [2a+ (n-1)d]

Or

The Sum of n terms of an Arithmetic Progression can also be given by Sn,

S

_{n }= n * [First term+ Last term]/2

**Proof for the sum of n terms in an AP**

Let’s consider the Generalized representation of Arithmetic Progression, the sum of all the terms in the above sequence is given as,

a, a+d, a+2d, a+3d, a+4d, ………. a+ (n-1)d

S

_{n }= (a+ a+ d+ a+ 2d+ a+ 3d+ a+ 4d+….. a+ (n-1)d) ⇢ (a)Now lets rewrite the the above equation in reverse order we get the equation as,

S

_{n}= (a + (n-1)d + a + (n-2)d + a + (n-3)d + ….. + a) ⇢ (b)In the next step, add the equation (a) with equation (b), after addition, the result is as follows,

2S

_{n}= (2a+ (n-1)d + 2a+ (n-1)d+…….. + 2a+ (n-1)d) (n terms)2S

_{n }= [2a + (n-1)d] × d

### What is the sum of first 30 natural numbers?

**Solution: **

First 30 natural numbers are 1 to 30. So, n = 30

From the above equation, it is known that, a =1, d = 2 – 1 = 1, and a

_{n }= 30Using the above equation of sum of n terms in an AP and substituting the values,

S

_{n}= 30/2 [2 × 1+ (30-1) × 1]

S_{n }= 15 [2 + 29]

S_{n}= 15 [31]

S_{n}= 465

So, The sum of 1 to 30 is 465.

**Similar Questions**

**Question 1:** **What is the total sum of 10 to 40?**

**Solution: **

From 10 to 40, there are total 31 numbers. So, n = 31

From the given statement, it is known that, a = 10, d = 11-10 = 1, and a

_{n}= 40Using the above equation of sum of n terms in a AP and substituting the values,

S

_{n}= n [a + a_{n}]/2

S_{n}= 31 [10 + 40]/2

S_{n}= 31 [25]

S_{n}= 775

So, The sum of the of 10 to 40 is 775.

**Question 2: What is the total sum of the first 10 terms of sequence 3, 6, 9, 12?**

**Solution: **

From the given statement, it is known that, a = 3, d = 6-3 = 3, and n = 10

Using the above equation of sum of n terms in a AP and substituting the values,

S

_{n}= 10/2 [2×3 + (10-1) × 3]

S_{n}= 5 [6 + 27]

S_{n}= 5 [33]

S_{n}= 165

So, The sum of the first 10 terms of given sequence is 165.

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