Question:
Find the average of odd numbers from 15 to 691
Correct Answer
353
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 691
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 691 are
15, 17, 19, . . . . 691
After observing the above list of the odd numbers from 15 to 691 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 691 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 691
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 691
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 691
= 15 + 691/2
= 706/2 = 353
Thus, the average of the odd numbers from 15 to 691 = 353 Answer
Method (2) to find the average of the odd numbers from 15 to 691
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 691 are
15, 17, 19, . . . . 691
The odd numbers from 15 to 691 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 691
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 691
691 = 15 + (n – 1) × 2
⇒ 691 = 15 + 2 n – 2
⇒ 691 = 15 – 2 + 2 n
⇒ 691 = 13 + 2 n
After transposing 13 to LHS
⇒ 691 – 13 = 2 n
⇒ 678 = 2 n
After rearranging the above expression
⇒ 2 n = 678
After transposing 2 to RHS
⇒ n = 678/2
⇒ n = 339
Thus, the number of terms of odd numbers from 15 to 691 = 339
This means 691 is the 339th term.
Finding the sum of the given odd numbers from 15 to 691
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 691
= 339/2 (15 + 691)
= 339/2 × 706
= 339 × 706/2
= 239334/2 = 119667
Thus, the sum of all terms of the given odd numbers from 15 to 691 = 119667
And, the total number of terms = 339
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 691
= 119667/339 = 353
Thus, the average of the given odd numbers from 15 to 691 = 353 Answer
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