Question : Find the average of odd numbers from 11 to 583
Correct Answer 297
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 583
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 583 are
11, 13, 15, . . . . 583
After observing the above list of the odd numbers from 11 to 583 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 583 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 583
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 583
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 583
= 11 + 583/2
= 594/2 = 297
Thus, the average of the odd numbers from 11 to 583 = 297 Answer
Method (2) to find the average of the odd numbers from 11 to 583
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 583 are
11, 13, 15, . . . . 583
The odd numbers from 11 to 583 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 583
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 583
583 = 11 + (n – 1) × 2
⇒ 583 = 11 + 2 n – 2
⇒ 583 = 11 – 2 + 2 n
⇒ 583 = 9 + 2 n
After transposing 9 to LHS
⇒ 583 – 9 = 2 n
⇒ 574 = 2 n
After rearranging the above expression
⇒ 2 n = 574
After transposing 2 to RHS
⇒ n = 574/2
⇒ n = 287
Thus, the number of terms of odd numbers from 11 to 583 = 287
This means 583 is the 287th term.
Finding the sum of the given odd numbers from 11 to 583
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 583
= 287/2 (11 + 583)
= 287/2 × 594
= 287 × 594/2
= 170478/2 = 85239
Thus, the sum of all terms of the given odd numbers from 11 to 583 = 85239
And, the total number of terms = 287
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 583
= 85239/287 = 297
Thus, the average of the given odd numbers from 11 to 583 = 297 Answer
Similar Questions
(1) Find the average of the first 1682 odd numbers.
(2) Find the average of odd numbers from 13 to 1397
(3) What will be the average of the first 4913 odd numbers?
(4) Find the average of even numbers from 12 to 1028
(5) Find the average of the first 698 odd numbers.
(6) Find the average of odd numbers from 3 to 219
(7) Find the average of odd numbers from 7 to 751
(8) Find the average of even numbers from 8 to 1402
(9) What is the average of the first 254 even numbers?
(10) Find the average of the first 2090 odd numbers.