Question:
Find the average of odd numbers from 3 to 599
Correct Answer
301
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 599
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 599 are
3, 5, 7, . . . . 599
After observing the above list of the odd numbers from 3 to 599 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 599 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 599
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 599
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 3 to 599
= 3 + 599/2
= 602/2 = 301
Thus, the average of the odd numbers from 3 to 599 = 301 Answer
Method (2) to find the average of the odd numbers from 3 to 599
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 599 are
3, 5, 7, . . . . 599
The odd numbers from 3 to 599 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 599
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 3 to 599
599 = 3 + (n – 1) × 2
⇒ 599 = 3 + 2 n – 2
⇒ 599 = 3 – 2 + 2 n
⇒ 599 = 1 + 2 n
After transposing 1 to LHS
⇒ 599 – 1 = 2 n
⇒ 598 = 2 n
After rearranging the above expression
⇒ 2 n = 598
After transposing 2 to RHS
⇒ n = 598/2
⇒ n = 299
Thus, the number of terms of odd numbers from 3 to 599 = 299
This means 599 is the 299th term.
Finding the sum of the given odd numbers from 3 to 599
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 3 to 599
= 299/2 (3 + 599)
= 299/2 × 602
= 299 × 602/2
= 179998/2 = 89999
Thus, the sum of all terms of the given odd numbers from 3 to 599 = 89999
And, the total number of terms = 299
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 3 to 599
= 89999/299 = 301
Thus, the average of the given odd numbers from 3 to 599 = 301 Answer
Similar Questions
(1) Find the average of even numbers from 4 to 1202
(2) What will be the average of the first 4308 odd numbers?
(3) Find the average of even numbers from 4 to 1524
(4) Find the average of even numbers from 6 to 110
(5) Find the average of odd numbers from 15 to 175
(6) Find the average of even numbers from 4 to 1976
(7) What is the average of the first 867 even numbers?
(8) Find the average of the first 3692 odd numbers.
(9) Find the average of even numbers from 4 to 1624
(10) Find the average of even numbers from 8 to 208