Question:
Find the average of odd numbers from 7 to 505
Correct Answer
256
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 505
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 505 are
7, 9, 11, . . . . 505
After observing the above list of the odd numbers from 7 to 505 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 505 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 505
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 505
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 7 to 505
= 7 + 505/2
= 512/2 = 256
Thus, the average of the odd numbers from 7 to 505 = 256 Answer
Method (2) to find the average of the odd numbers from 7 to 505
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 505 are
7, 9, 11, . . . . 505
The odd numbers from 7 to 505 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 505
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 7 to 505
505 = 7 + (n – 1) × 2
⇒ 505 = 7 + 2 n – 2
⇒ 505 = 7 – 2 + 2 n
⇒ 505 = 5 + 2 n
After transposing 5 to LHS
⇒ 505 – 5 = 2 n
⇒ 500 = 2 n
After rearranging the above expression
⇒ 2 n = 500
After transposing 2 to RHS
⇒ n = 500/2
⇒ n = 250
Thus, the number of terms of odd numbers from 7 to 505 = 250
This means 505 is the 250th term.
Finding the sum of the given odd numbers from 7 to 505
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 7 to 505
= 250/2 (7 + 505)
= 250/2 × 512
= 250 × 512/2
= 128000/2 = 64000
Thus, the sum of all terms of the given odd numbers from 7 to 505 = 64000
And, the total number of terms = 250
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 7 to 505
= 64000/250 = 256
Thus, the average of the given odd numbers from 7 to 505 = 256 Answer
Similar Questions
(1) Find the average of the first 2746 odd numbers.
(2) Find the average of the first 3007 even numbers.
(3) Find the average of even numbers from 10 to 1956
(4) What will be the average of the first 4698 odd numbers?
(5) What is the average of the first 1798 even numbers?
(6) Find the average of the first 3900 odd numbers.
(7) Find the average of even numbers from 6 to 1318
(8) Find the average of even numbers from 8 to 872
(9) What will be the average of the first 4164 odd numbers?
(10) Find the average of even numbers from 12 to 446