Question:
Find the average of odd numbers from 5 to 655
Correct Answer
330
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 655
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 655 are
5, 7, 9, . . . . 655
After observing the above list of the odd numbers from 5 to 655 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 655 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 655
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 655
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 655
= 5 + 655/2
= 660/2 = 330
Thus, the average of the odd numbers from 5 to 655 = 330 Answer
Method (2) to find the average of the odd numbers from 5 to 655
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 655 are
5, 7, 9, . . . . 655
The odd numbers from 5 to 655 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 655
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 655
655 = 5 + (n – 1) × 2
⇒ 655 = 5 + 2 n – 2
⇒ 655 = 5 – 2 + 2 n
⇒ 655 = 3 + 2 n
After transposing 3 to LHS
⇒ 655 – 3 = 2 n
⇒ 652 = 2 n
After rearranging the above expression
⇒ 2 n = 652
After transposing 2 to RHS
⇒ n = 652/2
⇒ n = 326
Thus, the number of terms of odd numbers from 5 to 655 = 326
This means 655 is the 326th term.
Finding the sum of the given odd numbers from 5 to 655
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 655
= 326/2 (5 + 655)
= 326/2 × 660
= 326 × 660/2
= 215160/2 = 107580
Thus, the sum of all terms of the given odd numbers from 5 to 655 = 107580
And, the total number of terms = 326
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 655
= 107580/326 = 330
Thus, the average of the given odd numbers from 5 to 655 = 330 Answer
Similar Questions
(1) What is the average of the first 1854 even numbers?
(2) Find the average of even numbers from 10 to 1510
(3) Find the average of odd numbers from 9 to 801
(4) Find the average of the first 1553 odd numbers.
(5) Find the average of odd numbers from 15 to 931
(6) Find the average of odd numbers from 7 to 249
(7) Find the average of the first 3899 even numbers.
(8) Find the average of odd numbers from 3 to 885
(9) What is the average of the first 1870 even numbers?
(10) Find the average of even numbers from 10 to 662