Question : Find the average of odd numbers from 9 to 527
Correct Answer 268
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 527
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 527 are
9, 11, 13, . . . . 527
After observing the above list of the odd numbers from 9 to 527 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 527 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 527
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 527
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 9 to 527
= 9 + 527/2
= 536/2 = 268
Thus, the average of the odd numbers from 9 to 527 = 268 Answer
Method (2) to find the average of the odd numbers from 9 to 527
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 527 are
9, 11, 13, . . . . 527
The odd numbers from 9 to 527 form an Arithmetic Series in which
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 527
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 9 to 527
527 = 9 + (n – 1) × 2
⇒ 527 = 9 + 2 n – 2
⇒ 527 = 9 – 2 + 2 n
⇒ 527 = 7 + 2 n
After transposing 7 to LHS
⇒ 527 – 7 = 2 n
⇒ 520 = 2 n
After rearranging the above expression
⇒ 2 n = 520
After transposing 2 to RHS
⇒ n = 520/2
⇒ n = 260
Thus, the number of terms of odd numbers from 9 to 527 = 260
This means 527 is the 260th term.
Finding the sum of the given odd numbers from 9 to 527
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 9 to 527
= 260/2 (9 + 527)
= 260/2 × 536
= 260 × 536/2
= 139360/2 = 69680
Thus, the sum of all terms of the given odd numbers from 9 to 527 = 69680
And, the total number of terms = 260
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 9 to 527
= 69680/260 = 268
Thus, the average of the given odd numbers from 9 to 527 = 268 Answer
Similar Questions
(1) What will be the average of the first 4914 odd numbers?
(2) What will be the average of the first 4448 odd numbers?
(3) Find the average of odd numbers from 5 to 1145
(4) Find the average of odd numbers from 13 to 1303
(5) Find the average of the first 2985 even numbers.
(6) Find the average of odd numbers from 5 to 989
(7) Find the average of the first 2844 odd numbers.
(8) Find the average of even numbers from 12 to 1768
(9) Find the average of even numbers from 6 to 1670
(10) What is the average of the first 239 even numbers?