Question:
Find the average of odd numbers from 13 to 745
Correct Answer
379
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 745
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 745 are
13, 15, 17, . . . . 745
After observing the above list of the odd numbers from 13 to 745 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 745 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 745
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 745
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 13 to 745
= 13 + 745/2
= 758/2 = 379
Thus, the average of the odd numbers from 13 to 745 = 379 Answer
Method (2) to find the average of the odd numbers from 13 to 745
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 745 are
13, 15, 17, . . . . 745
The odd numbers from 13 to 745 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 745
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 13 to 745
745 = 13 + (n – 1) × 2
⇒ 745 = 13 + 2 n – 2
⇒ 745 = 13 – 2 + 2 n
⇒ 745 = 11 + 2 n
After transposing 11 to LHS
⇒ 745 – 11 = 2 n
⇒ 734 = 2 n
After rearranging the above expression
⇒ 2 n = 734
After transposing 2 to RHS
⇒ n = 734/2
⇒ n = 367
Thus, the number of terms of odd numbers from 13 to 745 = 367
This means 745 is the 367th term.
Finding the sum of the given odd numbers from 13 to 745
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 13 to 745
= 367/2 (13 + 745)
= 367/2 × 758
= 367 × 758/2
= 278186/2 = 139093
Thus, the sum of all terms of the given odd numbers from 13 to 745 = 139093
And, the total number of terms = 367
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 13 to 745
= 139093/367 = 379
Thus, the average of the given odd numbers from 13 to 745 = 379 Answer
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