Question:
Find the average of odd numbers from 7 to 821
Correct Answer
414
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 821
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 821 are
7, 9, 11, . . . . 821
After observing the above list of the odd numbers from 7 to 821 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 821 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 821
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 821
The average of the numbers forming an Arithmetic Series
= The first term (a) + The last term (ℓ)/2
⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2
Thus, the average of the odd numbers from 7 to 821
= 7 + 821/2
= 828/2 = 414
Thus, the average of the odd numbers from 7 to 821 = 414 Answer
Method (2) to find the average of the odd numbers from 7 to 821
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 821 are
7, 9, 11, . . . . 821
The odd numbers from 7 to 821 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 821
The Average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers
Finding the number of terms
For an Arithmetic Series, the nth term
an = a + (n – 1) d
Where
a = First term
d = Common difference
n = number of terms
an = nth term
Thus, for the given series of the odd numbers from 7 to 821
821 = 7 + (n – 1) × 2
⇒ 821 = 7 + 2 n – 2
⇒ 821 = 7 – 2 + 2 n
⇒ 821 = 5 + 2 n
After transposing 5 to LHS
⇒ 821 – 5 = 2 n
⇒ 816 = 2 n
After rearranging the above expression
⇒ 2 n = 816
After transposing 2 to RHS
⇒ n = 816/2
⇒ n = 408
Thus, the number of terms of odd numbers from 7 to 821 = 408
This means 821 is the 408th term.
Finding the sum of the given odd numbers from 7 to 821
The sum of all terms (S) in an Arithmetic Series
= n/2 (a + ℓ)
Where, n = number of terms
a = First term
And, ℓ = Last term
Thus, the sum of all terms (S) of the given odd numbers from 7 to 821
= 408/2 (7 + 821)
= 408/2 × 828
= 408 × 828/2
= 337824/2 = 168912
Thus, the sum of all terms of the given odd numbers from 7 to 821 = 168912
And, the total number of terms = 408
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given odd numbers from 7 to 821
= 168912/408 = 414
Thus, the average of the given odd numbers from 7 to 821 = 414 Answer
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