Question : Find the average of odd numbers from 7 to 813
Correct Answer 410
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 813
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 813 are
7, 9, 11, . . . . 813
After observing the above list of the odd numbers from 7 to 813 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 813 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 813
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 813
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 813
= 7 + 813/2
= 820/2 = 410
Thus, the average of the odd numbers from 7 to 813 = 410 Answer
Method (2) to find the average of the odd numbers from 7 to 813
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 813 are
7, 9, 11, . . . . 813
The odd numbers from 7 to 813 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 813
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 813
813 = 7 + (n – 1) × 2
⇒ 813 = 7 + 2 n – 2
⇒ 813 = 7 – 2 + 2 n
⇒ 813 = 5 + 2 n
After transposing 5 to LHS
⇒ 813 – 5 = 2 n
⇒ 808 = 2 n
After rearranging the above expression
⇒ 2 n = 808
After transposing 2 to RHS
⇒ n = 808/2
⇒ n = 404
Thus, the number of terms of odd numbers from 7 to 813 = 404
This means 813 is the 404th term.
Finding the sum of the given odd numbers from 7 to 813
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 813
= 404/2 (7 + 813)
= 404/2 × 820
= 404 × 820/2
= 331280/2 = 165640
Thus, the sum of all terms of the given odd numbers from 7 to 813 = 165640
And, the total number of terms = 404
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 813
= 165640/404 = 410
Thus, the average of the given odd numbers from 7 to 813 = 410 Answer
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