Question : Find the average of odd numbers from 13 to 537
Correct Answer 275
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 537
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 537 are
13, 15, 17, . . . . 537
After observing the above list of the odd numbers from 13 to 537 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 537 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 537
The First Term (a) = 13
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 13 to 537
= 13 + 537/2
= 550/2 = 275
Thus, the average of the odd numbers from 13 to 537 = 275 Answer
Method (2) to find the average of the odd numbers from 13 to 537
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 537 are
13, 15, 17, . . . . 537
The odd numbers from 13 to 537 form an Arithmetic Series in which
The First Term (a) = 13
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 13 to 537
537 = 13 + (n – 1) × 2
⇒ 537 = 13 + 2 n – 2
⇒ 537 = 13 – 2 + 2 n
⇒ 537 = 11 + 2 n
After transposing 11 to LHS
⇒ 537 – 11 = 2 n
⇒ 526 = 2 n
After rearranging the above expression
⇒ 2 n = 526
After transposing 2 to RHS
⇒ n = 526/2
⇒ n = 263
Thus, the number of terms of odd numbers from 13 to 537 = 263
This means 537 is the 263th term.
Finding the sum of the given odd numbers from 13 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 13 to 537
= 263/2 (13 + 537)
= 263/2 × 550
= 263 × 550/2
= 144650/2 = 72325
Thus, the sum of all terms of the given odd numbers from 13 to 537 = 72325
And, the total number of terms = 263
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 537
= 72325/263 = 275
Thus, the average of the given odd numbers from 13 to 537 = 275 Answer
Similar Questions
(1) What is the average of the first 349 even numbers?
(2) Find the average of the first 3122 odd numbers.
(3) Find the average of the first 2649 even numbers.
(4) Find the average of even numbers from 4 to 484
(5) Find the average of the first 4544 even numbers.
(6) Find the average of the first 2218 odd numbers.
(7) Find the average of the first 4629 even numbers.
(8) Find the average of even numbers from 8 to 754
(9) Find the average of odd numbers from 13 to 261
(10) Find the average of the first 4652 even numbers.