Question

Find the sum of the first $30 $ natural numbers.

Answer 1

Hint: Natural numbers can be defined as the numbers which begin with the number one. Natural numbers are expressed as $1,2,3,4,.....$ difference between any two consecutive terms is always one and therefore since there is a common difference between the terms, here we will use arithmetic progression.

Complete step by step answer:
Given that to find the Sum of $1st\;{\text{30}}$ natural numbers.
The above word statement can be expressed as –
$1 + 2 + 3 + .... + 30$
Here, first term is $a = 1$
Common difference is $d = 3 - 2 = 2 - 1 = 1$
Also, here we have total number of terms, $n = 30$
Sum of terms in the arithmetic progression can be given by –
${S_n} = \dfrac{n}{2}[a + l]$
Place the known values in the above expression –
${S_{30}} = \dfrac{{30}}{2}[1 + 30]$
Simplify the above expression –
${S_{30}} = \dfrac{{30}}{2}[31]$
Find the factors for the term in the numerator of the fraction in the above expression –
${S_{30}} = \dfrac{{2 \times 15}}{2}[31]$
Common term from the numerator and the denominator cancels each other and therefore remove from the numerator and the denominator of the above expression –
${S_{30}} = 15[31]$
Simplify the above expression finding the product of the terms.
${S_{30}} = 465$
Hence, the sum of $1st\;{\text{30}}$ natural numbers is $465$.

Note: Always remember and know the concepts very clearly for the natural numbers, whole numbers, integers, natural numbers, prime numbers, the composite numbers, Rational and irrational numbers. Also, know the difference and co-relation between them. Also, remember the difference between the sequences and patterns of the series. When there is a common ratio between the terms then the series in the form of geometric progression and when there is a common difference between the consecutive terms then the series in the form of the arithmetic progression.