Question:
Find the average of odd numbers from 5 to 591
Correct Answer
298
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 591
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 591 are
5, 7, 9, . . . . 591
After observing the above list of the odd numbers from 5 to 591 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 591 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 591
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 591
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 591
= 5 + 591/2
= 596/2 = 298
Thus, the average of the odd numbers from 5 to 591 = 298 Answer
Method (2) to find the average of the odd numbers from 5 to 591
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 591 are
5, 7, 9, . . . . 591
The odd numbers from 5 to 591 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 591
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 591
591 = 5 + (n – 1) × 2
⇒ 591 = 5 + 2 n – 2
⇒ 591 = 5 – 2 + 2 n
⇒ 591 = 3 + 2 n
After transposing 3 to LHS
⇒ 591 – 3 = 2 n
⇒ 588 = 2 n
After rearranging the above expression
⇒ 2 n = 588
After transposing 2 to RHS
⇒ n = 588/2
⇒ n = 294
Thus, the number of terms of odd numbers from 5 to 591 = 294
This means 591 is the 294th term.
Finding the sum of the given odd numbers from 5 to 591
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 591
= 294/2 (5 + 591)
= 294/2 × 596
= 294 × 596/2
= 175224/2 = 87612
Thus, the sum of all terms of the given odd numbers from 5 to 591 = 87612
And, the total number of terms = 294
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 591
= 87612/294 = 298
Thus, the average of the given odd numbers from 5 to 591 = 298 Answer
Similar Questions
(1) What will be the average of the first 4638 odd numbers?
(2) Find the average of odd numbers from 7 to 315
(3) Find the average of odd numbers from 9 to 1073
(4) Find the average of even numbers from 4 to 1996
(5) Find the average of even numbers from 12 to 52
(6) Find the average of even numbers from 4 to 24
(7) Find the average of odd numbers from 13 to 27
(8) Find the average of odd numbers from 3 to 1449
(9) Find the average of even numbers from 8 to 662
(10) What is the average of the first 484 even numbers?