Question : Find the average of odd numbers from 13 to 499
Correct Answer 256
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 499
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 499 are
13, 15, 17, . . . . 499
After observing the above list of the odd numbers from 13 to 499 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 499 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 499
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 499
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 13 to 499
= 13 + 499/2
= 512/2 = 256
Thus, the average of the odd numbers from 13 to 499 = 256 Answer
Method (2) to find the average of the odd numbers from 13 to 499
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 499 are
13, 15, 17, . . . . 499
The odd numbers from 13 to 499 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 499
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 13 to 499
499 = 13 + (n – 1) × 2
⇒ 499 = 13 + 2 n – 2
⇒ 499 = 13 – 2 + 2 n
⇒ 499 = 11 + 2 n
After transposing 11 to LHS
⇒ 499 – 11 = 2 n
⇒ 488 = 2 n
After rearranging the above expression
⇒ 2 n = 488
After transposing 2 to RHS
⇒ n = 488/2
⇒ n = 244
Thus, the number of terms of odd numbers from 13 to 499 = 244
This means 499 is the 244th term.
Finding the sum of the given odd numbers from 13 to 499
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 13 to 499
= 244/2 (13 + 499)
= 244/2 × 512
= 244 × 512/2
= 124928/2 = 62464
Thus, the sum of all terms of the given odd numbers from 13 to 499 = 62464
And, the total number of terms = 244
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 13 to 499
= 62464/244 = 256
Thus, the average of the given odd numbers from 13 to 499 = 256 Answer
Similar Questions
(1) Find the average of the first 3859 odd numbers.
(2) Find the average of even numbers from 10 to 1618
(3) Find the average of the first 2779 even numbers.
(4) Find the average of the first 4964 even numbers.
(5) Find the average of the first 4629 even numbers.
(6) Find the average of the first 2210 even numbers.
(7) Find the average of even numbers from 4 to 80
(8) Find the average of odd numbers from 3 to 1277
(9) What is the average of the first 648 even numbers?
(10) What will be the average of the first 4283 odd numbers?