Question : Find the average of odd numbers from 15 to 739
Correct Answer 377
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 739
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 739 are
15, 17, 19, . . . . 739
After observing the above list of the odd numbers from 15 to 739 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 739 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 739
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 739
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 739
= 15 + 739/2
= 754/2 = 377
Thus, the average of the odd numbers from 15 to 739 = 377 Answer
Method (2) to find the average of the odd numbers from 15 to 739
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 739 are
15, 17, 19, . . . . 739
The odd numbers from 15 to 739 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 739
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 739
739 = 15 + (n – 1) × 2
⇒ 739 = 15 + 2 n – 2
⇒ 739 = 15 – 2 + 2 n
⇒ 739 = 13 + 2 n
After transposing 13 to LHS
⇒ 739 – 13 = 2 n
⇒ 726 = 2 n
After rearranging the above expression
⇒ 2 n = 726
After transposing 2 to RHS
⇒ n = 726/2
⇒ n = 363
Thus, the number of terms of odd numbers from 15 to 739 = 363
This means 739 is the 363th term.
Finding the sum of the given odd numbers from 15 to 739
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 739
= 363/2 (15 + 739)
= 363/2 × 754
= 363 × 754/2
= 273702/2 = 136851
Thus, the sum of all terms of the given odd numbers from 15 to 739 = 136851
And, the total number of terms = 363
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 739
= 136851/363 = 377
Thus, the average of the given odd numbers from 15 to 739 = 377 Answer
Similar Questions
(1) Find the average of the first 3832 even numbers.
(2) What is the average of the first 43 odd numbers?
(3) Find the average of the first 3661 even numbers.
(4) Find the average of odd numbers from 7 to 313
(5) Find the average of the first 4713 even numbers.
(6) Find the average of even numbers from 12 to 1640
(7) Find the average of even numbers from 10 to 500
(8) Find the average of even numbers from 10 to 1002
(9) What is the average of the first 679 even numbers?
(10) Find the average of odd numbers from 5 to 161