Question:
Find the average of odd numbers from 3 to 201
Correct Answer
102
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 201
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 201 are
3, 5, 7, . . . . 201
After observing the above list of the odd numbers from 3 to 201 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 201 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 201
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 201
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 201
= 3 + 201/2
= 204/2 = 102
Thus, the average of the odd numbers from 3 to 201 = 102 Answer
Method (2) to find the average of the odd numbers from 3 to 201
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 201 are
3, 5, 7, . . . . 201
The odd numbers from 3 to 201 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 201
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 201
201 = 3 + (n – 1) × 2
⇒ 201 = 3 + 2 n – 2
⇒ 201 = 3 – 2 + 2 n
⇒ 201 = 1 + 2 n
After transposing 1 to LHS
⇒ 201 – 1 = 2 n
⇒ 200 = 2 n
After rearranging the above expression
⇒ 2 n = 200
After transposing 2 to RHS
⇒ n = 200/2
⇒ n = 100
Thus, the number of terms of odd numbers from 3 to 201 = 100
This means 201 is the 100th term.
Finding the sum of the given odd numbers from 3 to 201
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 201
= 100/2 (3 + 201)
= 100/2 × 204
= 100 × 204/2
= 20400/2 = 10200
Thus, the sum of all terms of the given odd numbers from 3 to 201 = 10200
And, the total number of terms = 100
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 201
= 10200/100 = 102
Thus, the average of the given odd numbers from 3 to 201 = 102 Answer
Similar Questions
(1) What is the average of the first 1230 even numbers?
(2) Find the average of even numbers from 12 to 1540
(3) Find the average of odd numbers from 5 to 1077
(4) Find the average of odd numbers from 13 to 725
(5) Find the average of odd numbers from 15 to 1231
(6) Find the average of the first 470 odd numbers.
(7) Find the average of odd numbers from 11 to 327
(8) Find the average of the first 4450 even numbers.
(9) Find the average of odd numbers from 15 to 1163
(10) What is the average of the first 941 even numbers?