Question:
Find the average of odd numbers from 5 to 781
Correct Answer
393
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 781
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 781 are
5, 7, 9, . . . . 781
After observing the above list of the odd numbers from 5 to 781 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 781 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 781
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 781
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 5 to 781
= 5 + 781/2
= 786/2 = 393
Thus, the average of the odd numbers from 5 to 781 = 393 Answer
Method (2) to find the average of the odd numbers from 5 to 781
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 781 are
5, 7, 9, . . . . 781
The odd numbers from 5 to 781 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 781
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 5 to 781
781 = 5 + (n – 1) × 2
⇒ 781 = 5 + 2 n – 2
⇒ 781 = 5 – 2 + 2 n
⇒ 781 = 3 + 2 n
After transposing 3 to LHS
⇒ 781 – 3 = 2 n
⇒ 778 = 2 n
After rearranging the above expression
⇒ 2 n = 778
After transposing 2 to RHS
⇒ n = 778/2
⇒ n = 389
Thus, the number of terms of odd numbers from 5 to 781 = 389
This means 781 is the 389th term.
Finding the sum of the given odd numbers from 5 to 781
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 5 to 781
= 389/2 (5 + 781)
= 389/2 × 786
= 389 × 786/2
= 305754/2 = 152877
Thus, the sum of all terms of the given odd numbers from 5 to 781 = 152877
And, the total number of terms = 389
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 5 to 781
= 152877/389 = 393
Thus, the average of the given odd numbers from 5 to 781 = 393 Answer
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