Question:
Find the average of odd numbers from 13 to 427
Correct Answer
220
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 427
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 427 are
13, 15, 17, . . . . 427
After observing the above list of the odd numbers from 13 to 427 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 427 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 427
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 427
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 13 to 427
= 13 + 427/2
= 440/2 = 220
Thus, the average of the odd numbers from 13 to 427 = 220 Answer
Method (2) to find the average of the odd numbers from 13 to 427
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 427 are
13, 15, 17, . . . . 427
The odd numbers from 13 to 427 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 427
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 13 to 427
427 = 13 + (n – 1) × 2
⇒ 427 = 13 + 2 n – 2
⇒ 427 = 13 – 2 + 2 n
⇒ 427 = 11 + 2 n
After transposing 11 to LHS
⇒ 427 – 11 = 2 n
⇒ 416 = 2 n
After rearranging the above expression
⇒ 2 n = 416
After transposing 2 to RHS
⇒ n = 416/2
⇒ n = 208
Thus, the number of terms of odd numbers from 13 to 427 = 208
This means 427 is the 208th term.
Finding the sum of the given odd numbers from 13 to 427
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 13 to 427
= 208/2 (13 + 427)
= 208/2 × 440
= 208 × 440/2
= 91520/2 = 45760
Thus, the sum of all terms of the given odd numbers from 13 to 427 = 45760
And, the total number of terms = 208
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 13 to 427
= 45760/208 = 220
Thus, the average of the given odd numbers from 13 to 427 = 220 Answer
Similar Questions
(1) Find the average of even numbers from 4 to 848
(2) Find the average of even numbers from 12 to 214
(3) Find the average of even numbers from 6 to 904
(4) Find the average of even numbers from 8 to 492
(5) Find the average of the first 2629 even numbers.
(6) What will be the average of the first 4518 odd numbers?
(7) Find the average of the first 2073 even numbers.
(8) Find the average of the first 3873 odd numbers.
(9) Find the average of even numbers from 4 to 1744
(10) Find the average of the first 3828 odd numbers.