Question : Find the average of odd numbers from 15 to 813
Correct Answer 414
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 813
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 813 are
15, 17, 19, . . . . 813
After observing the above list of the odd numbers from 15 to 813 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 813 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 813
The First Term (a) = 15
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 15 to 813
= 15 + 813/2
= 828/2 = 414
Thus, the average of the odd numbers from 15 to 813 = 414 Answer
Method (2) to find the average of the odd numbers from 15 to 813
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 813 are
15, 17, 19, . . . . 813
The odd numbers from 15 to 813 form an Arithmetic Series in which
The First Term (a) = 15
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 15 to 813
813 = 15 + (n – 1) × 2
⇒ 813 = 15 + 2 n – 2
⇒ 813 = 15 – 2 + 2 n
⇒ 813 = 13 + 2 n
After transposing 13 to LHS
⇒ 813 – 13 = 2 n
⇒ 800 = 2 n
After rearranging the above expression
⇒ 2 n = 800
After transposing 2 to RHS
⇒ n = 800/2
⇒ n = 400
Thus, the number of terms of odd numbers from 15 to 813 = 400
This means 813 is the 400th term.
Finding the sum of the given odd numbers from 15 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 15 to 813
= 400/2 (15 + 813)
= 400/2 × 828
= 400 × 828/2
= 331200/2 = 165600
Thus, the sum of all terms of the given odd numbers from 15 to 813 = 165600
And, the total number of terms = 400
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 15 to 813
= 165600/400 = 414
Thus, the average of the given odd numbers from 15 to 813 = 414 Answer
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