Question : Find the average of odd numbers from 11 to 537
Correct Answer 274
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 537
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 537 are
11, 13, 15, . . . . 537
After observing the above list of the odd numbers from 11 to 537 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 537 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 537
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 537
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 11 to 537
= 11 + 537/2
= 548/2 = 274
Thus, the average of the odd numbers from 11 to 537 = 274 Answer
Method (2) to find the average of the odd numbers from 11 to 537
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 537 are
11, 13, 15, . . . . 537
The odd numbers from 11 to 537 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 537
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 11 to 537
537 = 11 + (n – 1) × 2
⇒ 537 = 11 + 2 n – 2
⇒ 537 = 11 – 2 + 2 n
⇒ 537 = 9 + 2 n
After transposing 9 to LHS
⇒ 537 – 9 = 2 n
⇒ 528 = 2 n
After rearranging the above expression
⇒ 2 n = 528
After transposing 2 to RHS
⇒ n = 528/2
⇒ n = 264
Thus, the number of terms of odd numbers from 11 to 537 = 264
This means 537 is the 264th term.
Finding the sum of the given odd numbers from 11 to 537
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 11 to 537
= 264/2 (11 + 537)
= 264/2 × 548
= 264 × 548/2
= 144672/2 = 72336
Thus, the sum of all terms of the given odd numbers from 11 to 537 = 72336
And, the total number of terms = 264
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 11 to 537
= 72336/264 = 274
Thus, the average of the given odd numbers from 11 to 537 = 274 Answer
Similar Questions
(1) What will be the average of the first 4661 odd numbers?
(2) Find the average of odd numbers from 15 to 691
(3) Find the average of even numbers from 6 to 1594
(4) Find the average of odd numbers from 15 to 717
(5) Find the average of odd numbers from 5 to 543
(6) Find the average of the first 4337 even numbers.
(7) Find the average of even numbers from 12 to 1560
(8) What is the average of the first 1821 even numbers?
(9) Find the average of odd numbers from 13 to 541
(10) Find the average of the first 4863 even numbers.