Question:
Find the average of odd numbers from 9 to 371
Correct Answer
190
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 371
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 371 are
9, 11, 13, . . . . 371
After observing the above list of the odd numbers from 9 to 371 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 371 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 371
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 371
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 9 to 371
= 9 + 371/2
= 380/2 = 190
Thus, the average of the odd numbers from 9 to 371 = 190 Answer
Method (2) to find the average of the odd numbers from 9 to 371
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 371 are
9, 11, 13, . . . . 371
The odd numbers from 9 to 371 form an Arithmetic Series in which
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 371
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 9 to 371
371 = 9 + (n – 1) × 2
⇒ 371 = 9 + 2 n – 2
⇒ 371 = 9 – 2 + 2 n
⇒ 371 = 7 + 2 n
After transposing 7 to LHS
⇒ 371 – 7 = 2 n
⇒ 364 = 2 n
After rearranging the above expression
⇒ 2 n = 364
After transposing 2 to RHS
⇒ n = 364/2
⇒ n = 182
Thus, the number of terms of odd numbers from 9 to 371 = 182
This means 371 is the 182th term.
Finding the sum of the given odd numbers from 9 to 371
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 9 to 371
= 182/2 (9 + 371)
= 182/2 × 380
= 182 × 380/2
= 69160/2 = 34580
Thus, the sum of all terms of the given odd numbers from 9 to 371 = 34580
And, the total number of terms = 182
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 9 to 371
= 34580/182 = 190
Thus, the average of the given odd numbers from 9 to 371 = 190 Answer
Similar Questions
(1) Find the average of the first 2219 even numbers.
(2) Find the average of even numbers from 10 to 550
(3) Find the average of the first 492 odd numbers.
(4) Find the average of the first 3359 odd numbers.
(5) Find the average of the first 1678 odd numbers.
(6) Find the average of the first 671 odd numbers.
(7) What will be the average of the first 4068 odd numbers?
(8) Find the average of the first 790 odd numbers.
(9) Find the average of the first 915 odd numbers.
(10) Find the average of the first 4777 even numbers.