Sum of certain consecutive odd positive numbers is ${{57}^{2}}-{{13}^{2}}$ . Find them.
Last updated date: 17th Mar 2023
•
Total views: 304.2k
•
Views today: 6.83k
Answer
304.2k+ views
Hint: The question is related to the sum of consecutive odd positive numbers. The sum of first $n$ consecutive odd positive numbers is equal to ${{n}^{2}}$ . So, the given expression represents the sum of odd numbers greater than $13$ and less than or equal to $57$ .
Complete step-by-step answer:
The series of consecutive odd numbers is an arithmetic progression with common difference $2$ . If the first $n$ odd numbers are considered then, it is an arithmetic progression with the first term equal to $1$ , common difference equal to $2$ , and the number of terms equal to $n$ . Now, we know , the sum of first $n$ terms of an arithmetic progression with the firth term equal to $a$ and the common difference equal to $d$ is given as ${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$ . So, the sum of first $n$ consecutive odd positive numbers will be equal to $\dfrac{n}{2}\left[ 2+\left( n-1 \right)2 \right]$ .
\[=\dfrac{n}{2}\left[ 2+2n-2 \right]\]
\[=\dfrac{n}{2}\left[ 2n \right]\]
$={{n}^{2}}$
So, the sum of first $n$ consecutive odd positive numbers is equal to ${{n}^{2}}$ .
Now, coming to the question, we are given the expression ${{57}^{2}}-{{13}^{2}}$. If we analyze the expression carefully, we can conclude that it represents the difference of the sum of first $57$ consecutive odd positive numbers and first $13$ consecutive odd positive numbers. So, the expression ${{57}^{2}}-{{13}^{2}}$ represents the sum of odd numbers greater than $13$ and less than or equal to $57$ . So, the numbers are $15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55$ and $57$.
Note: The sum of first $n$ natural numbers is equal to $\dfrac{n\left( n+1 \right)}{2}$ . The sum of first $n$ even natural numbers is equal to $\dfrac{n\left( n+2 \right)}{2}$ and the sum of first $n$ odd natural numbers is equal to ${{n}^{2}}$ . These formulae should be remembered as they are frequently used and there should be no confusion between the formulae.
Complete step-by-step answer:
The series of consecutive odd numbers is an arithmetic progression with common difference $2$ . If the first $n$ odd numbers are considered then, it is an arithmetic progression with the first term equal to $1$ , common difference equal to $2$ , and the number of terms equal to $n$ . Now, we know , the sum of first $n$ terms of an arithmetic progression with the firth term equal to $a$ and the common difference equal to $d$ is given as ${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$ . So, the sum of first $n$ consecutive odd positive numbers will be equal to $\dfrac{n}{2}\left[ 2+\left( n-1 \right)2 \right]$ .
\[=\dfrac{n}{2}\left[ 2+2n-2 \right]\]
\[=\dfrac{n}{2}\left[ 2n \right]\]
$={{n}^{2}}$
So, the sum of first $n$ consecutive odd positive numbers is equal to ${{n}^{2}}$ .
Now, coming to the question, we are given the expression ${{57}^{2}}-{{13}^{2}}$. If we analyze the expression carefully, we can conclude that it represents the difference of the sum of first $57$ consecutive odd positive numbers and first $13$ consecutive odd positive numbers. So, the expression ${{57}^{2}}-{{13}^{2}}$ represents the sum of odd numbers greater than $13$ and less than or equal to $57$ . So, the numbers are $15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55$ and $57$.
Note: The sum of first $n$ natural numbers is equal to $\dfrac{n\left( n+1 \right)}{2}$ . The sum of first $n$ even natural numbers is equal to $\dfrac{n\left( n+2 \right)}{2}$ and the sum of first $n$ odd natural numbers is equal to ${{n}^{2}}$ . These formulae should be remembered as they are frequently used and there should be no confusion between the formulae.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
