Question : Find the average of odd numbers from 5 to 587
Correct Answer 296
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 587
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 587 are
5, 7, 9, . . . . 587
After observing the above list of the odd numbers from 5 to 587 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 587 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 587
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 587
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 5 to 587
= 5 + 587/2
= 592/2 = 296
Thus, the average of the odd numbers from 5 to 587 = 296 Answer
Method (2) to find the average of the odd numbers from 5 to 587
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 587 are
5, 7, 9, . . . . 587
The odd numbers from 5 to 587 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 587
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 5 to 587
587 = 5 + (n – 1) × 2
⇒ 587 = 5 + 2 n – 2
⇒ 587 = 5 – 2 + 2 n
⇒ 587 = 3 + 2 n
After transposing 3 to LHS
⇒ 587 – 3 = 2 n
⇒ 584 = 2 n
After rearranging the above expression
⇒ 2 n = 584
After transposing 2 to RHS
⇒ n = 584/2
⇒ n = 292
Thus, the number of terms of odd numbers from 5 to 587 = 292
This means 587 is the 292th term.
Finding the sum of the given odd numbers from 5 to 587
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 5 to 587
= 292/2 (5 + 587)
= 292/2 × 592
= 292 × 592/2
= 172864/2 = 86432
Thus, the sum of all terms of the given odd numbers from 5 to 587 = 86432
And, the total number of terms = 292
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 5 to 587
= 86432/292 = 296
Thus, the average of the given odd numbers from 5 to 587 = 296 Answer
Similar Questions
(1) Find the average of the first 2821 even numbers.
(2) Find the average of even numbers from 8 to 1078
(3) Find the average of the first 4108 even numbers.
(4) Find the average of even numbers from 8 to 432
(5) Find the average of odd numbers from 9 to 925
(6) What is the average of the first 1923 even numbers?
(7) What is the average of the first 833 even numbers?
(8) Find the average of odd numbers from 9 to 677
(9) Find the average of odd numbers from 15 to 1467
(10) Find the average of the first 386 odd numbers.