Question : Find the average of odd numbers from 5 to 625
Correct Answer 315
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 625
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 625 are
5, 7, 9, . . . . 625
After observing the above list of the odd numbers from 5 to 625 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 625 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 625
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 625
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 625
= 5 + 625/2
= 630/2 = 315
Thus, the average of the odd numbers from 5 to 625 = 315 Answer
Method (2) to find the average of the odd numbers from 5 to 625
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 625 are
5, 7, 9, . . . . 625
The odd numbers from 5 to 625 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 625
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 625
625 = 5 + (n – 1) × 2
⇒ 625 = 5 + 2 n – 2
⇒ 625 = 5 – 2 + 2 n
⇒ 625 = 3 + 2 n
After transposing 3 to LHS
⇒ 625 – 3 = 2 n
⇒ 622 = 2 n
After rearranging the above expression
⇒ 2 n = 622
After transposing 2 to RHS
⇒ n = 622/2
⇒ n = 311
Thus, the number of terms of odd numbers from 5 to 625 = 311
This means 625 is the 311th term.
Finding the sum of the given odd numbers from 5 to 625
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 625
= 311/2 (5 + 625)
= 311/2 × 630
= 311 × 630/2
= 195930/2 = 97965
Thus, the sum of all terms of the given odd numbers from 5 to 625 = 97965
And, the total number of terms = 311
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 625
= 97965/311 = 315
Thus, the average of the given odd numbers from 5 to 625 = 315 Answer
Similar Questions
(1) Find the average of odd numbers from 11 to 1279
(2) Find the average of the first 314 odd numbers.
(3) Find the average of the first 3409 even numbers.
(4) Find the average of odd numbers from 15 to 1703
(5) Find the average of odd numbers from 7 to 513
(6) Find the average of odd numbers from 15 to 345
(7) What is the average of the first 1844 even numbers?
(8) Find the average of the first 3750 odd numbers.
(9) Find the average of the first 1402 odd numbers.
(10) Find the average of odd numbers from 11 to 829