Question:
Find the average of odd numbers from 5 to 255
Correct Answer
130
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 255
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 255 are
5, 7, 9, . . . . 255
After observing the above list of the odd numbers from 5 to 255 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 255 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 255
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 255
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 255
= 5 + 255/2
= 260/2 = 130
Thus, the average of the odd numbers from 5 to 255 = 130 Answer
Method (2) to find the average of the odd numbers from 5 to 255
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 255 are
5, 7, 9, . . . . 255
The odd numbers from 5 to 255 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 255
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 255
255 = 5 + (n – 1) × 2
⇒ 255 = 5 + 2 n – 2
⇒ 255 = 5 – 2 + 2 n
⇒ 255 = 3 + 2 n
After transposing 3 to LHS
⇒ 255 – 3 = 2 n
⇒ 252 = 2 n
After rearranging the above expression
⇒ 2 n = 252
After transposing 2 to RHS
⇒ n = 252/2
⇒ n = 126
Thus, the number of terms of odd numbers from 5 to 255 = 126
This means 255 is the 126th term.
Finding the sum of the given odd numbers from 5 to 255
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 255
= 126/2 (5 + 255)
= 126/2 × 260
= 126 × 260/2
= 32760/2 = 16380
Thus, the sum of all terms of the given odd numbers from 5 to 255 = 16380
And, the total number of terms = 126
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 255
= 16380/126 = 130
Thus, the average of the given odd numbers from 5 to 255 = 130 Answer
Similar Questions
(1) Find the average of odd numbers from 11 to 565
(2) Find the average of odd numbers from 13 to 1259
(3) Find the average of the first 4547 even numbers.
(4) Find the average of odd numbers from 3 to 667
(5) Find the average of even numbers from 12 to 1742
(6) Find the average of the first 3157 odd numbers.
(7) Find the average of the first 4479 even numbers.
(8) Find the average of the first 2537 even numbers.
(9) Find the average of the first 4495 even numbers.
(10) Find the average of even numbers from 6 to 1222