Question:
Find the average of odd numbers from 3 to 579
Correct Answer
291
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 579
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 579 are
3, 5, 7, . . . . 579
After observing the above list of the odd numbers from 3 to 579 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 579 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 579
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 579
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 3 to 579
= 3 + 579/2
= 582/2 = 291
Thus, the average of the odd numbers from 3 to 579 = 291 Answer
Method (2) to find the average of the odd numbers from 3 to 579
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 579 are
3, 5, 7, . . . . 579
The odd numbers from 3 to 579 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 579
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 3 to 579
579 = 3 + (n – 1) × 2
⇒ 579 = 3 + 2 n – 2
⇒ 579 = 3 – 2 + 2 n
⇒ 579 = 1 + 2 n
After transposing 1 to LHS
⇒ 579 – 1 = 2 n
⇒ 578 = 2 n
After rearranging the above expression
⇒ 2 n = 578
After transposing 2 to RHS
⇒ n = 578/2
⇒ n = 289
Thus, the number of terms of odd numbers from 3 to 579 = 289
This means 579 is the 289th term.
Finding the sum of the given odd numbers from 3 to 579
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 3 to 579
= 289/2 (3 + 579)
= 289/2 × 582
= 289 × 582/2
= 168198/2 = 84099
Thus, the sum of all terms of the given odd numbers from 3 to 579 = 84099
And, the total number of terms = 289
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 3 to 579
= 84099/289 = 291
Thus, the average of the given odd numbers from 3 to 579 = 291 Answer
Similar Questions
(1) What is the average of the first 118 even numbers?
(2) What is the average of the first 128 odd numbers?
(3) Find the average of the first 4410 even numbers.
(4) Find the average of even numbers from 6 to 436
(5) Find the average of odd numbers from 9 to 683
(6) Find the average of the first 4356 even numbers.
(7) Find the average of even numbers from 12 to 1418
(8) Find the average of even numbers from 12 to 1404
(9) Find the average of the first 423 odd numbers.
(10) What is the average of the first 35 odd numbers?