Question

Prathyusha stated that “the average of the first 10 odd numbers is also 10. “ Do you agree with her? Justify your answer.

Answer 1

Hint: Here we may find the average of the first ten odd numbers and will check whether it equals to 10 or not. While finding the average, we will be using the formula for finding the sum of the given terms of an A.P.

Complete step-by-step answer:
Since, we know that the first ten odd numbers are: - 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.
So, we may use here the formula of average to find the average of these 10 numbers and the formula for average is given as:
\[Avg=\dfrac{Sum\,\,\,of\,numbers}{Total\,number\,\,of\,numbers}............\left( 1 \right)\]
Now, for finding the sum of these 10 odd numbers, we can easily see that the sequence of these numbers forms an A.P.
In this A.P., the first term is = 1
And the common difference is:
d= 3-1 = 2
So we have an AP whose first term is 1 and the common difference is 2.
Now for finding the sum of first 10 terms of this AP, we may use the formula for sum of n terms of an AP, which is given as:
${{S}_{n}}=\dfrac{n}{2}\left\{ 2a+\left( n-1 \right)d \right\}$............(1)
Now on substituting the values of a, n, and d in equation (1) we get:
${{S}_{10}}=\dfrac{10}{2}\left\{ 2\times 1+\left( 10-1 \right)\times 2 \right\}$
 ${{S}_{10}}=5\left( 2+9\times 2 \right)$
 ${{S}_{10}}=5\left( 2+18 \right)$
 ${{S}_{10}}=5\times 20$
 ${{S}_{10}}=100$
Therefore, we get that the sum of the first ten odd numbers is 100.
Now on substituting this value of sum in equation (1) to find the average, we get:
$Avg=\dfrac{100}{10}$
 $Avg=10$
So, the average of the first 10 odd numbers is 10.
Hence, Pratyusha is right with his statement.

Note: Here we can also use another formula for the sum of terms of an AP which is ${{S}_{n}}=\dfrac{n}{2}$( first term + last term). As here we know both the first term and the last term, so using this formula will be a shortcut to get the sum.