Question : Find the average of odd numbers from 5 to 273
Correct Answer 139
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 273
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 273 are
5, 7, 9, . . . . 273
After observing the above list of the odd numbers from 5 to 273 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 273 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 273
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 273
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 273
= 5 + 273/2
= 278/2 = 139
Thus, the average of the odd numbers from 5 to 273 = 139 Answer
Method (2) to find the average of the odd numbers from 5 to 273
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 273 are
5, 7, 9, . . . . 273
The odd numbers from 5 to 273 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 273
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 273
273 = 5 + (n – 1) × 2
⇒ 273 = 5 + 2 n – 2
⇒ 273 = 5 – 2 + 2 n
⇒ 273 = 3 + 2 n
After transposing 3 to LHS
⇒ 273 – 3 = 2 n
⇒ 270 = 2 n
After rearranging the above expression
⇒ 2 n = 270
After transposing 2 to RHS
⇒ n = 270/2
⇒ n = 135
Thus, the number of terms of odd numbers from 5 to 273 = 135
This means 273 is the 135th term.
Finding the sum of the given odd numbers from 5 to 273
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 273
= 135/2 (5 + 273)
= 135/2 × 278
= 135 × 278/2
= 37530/2 = 18765
Thus, the sum of all terms of the given odd numbers from 5 to 273 = 18765
And, the total number of terms = 135
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 273
= 18765/135 = 139
Thus, the average of the given odd numbers from 5 to 273 = 139 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 129
(2) Find the average of odd numbers from 13 to 277
(3) What is the average of the first 896 even numbers?
(4) Find the average of odd numbers from 13 to 889
(5) Find the average of odd numbers from 15 to 1283
(6) Find the average of even numbers from 6 to 224
(7) Find the average of the first 689 odd numbers.
(8) Find the average of the first 1440 odd numbers.
(9) Find the average of the first 1785 odd numbers.
(10) Find the average of odd numbers from 9 to 525