Question:
Find the average of odd numbers from 11 to 281
Correct Answer
146
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 281
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 281 are
11, 13, 15, . . . . 281
After observing the above list of the odd numbers from 11 to 281 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 281 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 281
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 281
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 11 to 281
= 11 + 281/2
= 292/2 = 146
Thus, the average of the odd numbers from 11 to 281 = 146 Answer
Method (2) to find the average of the odd numbers from 11 to 281
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 281 are
11, 13, 15, . . . . 281
The odd numbers from 11 to 281 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 281
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 11 to 281
281 = 11 + (n – 1) × 2
⇒ 281 = 11 + 2 n – 2
⇒ 281 = 11 – 2 + 2 n
⇒ 281 = 9 + 2 n
After transposing 9 to LHS
⇒ 281 – 9 = 2 n
⇒ 272 = 2 n
After rearranging the above expression
⇒ 2 n = 272
After transposing 2 to RHS
⇒ n = 272/2
⇒ n = 136
Thus, the number of terms of odd numbers from 11 to 281 = 136
This means 281 is the 136th term.
Finding the sum of the given odd numbers from 11 to 281
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 11 to 281
= 136/2 (11 + 281)
= 136/2 × 292
= 136 × 292/2
= 39712/2 = 19856
Thus, the sum of all terms of the given odd numbers from 11 to 281 = 19856
And, the total number of terms = 136
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 11 to 281
= 19856/136 = 146
Thus, the average of the given odd numbers from 11 to 281 = 146 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 1749
(2) Find the average of the first 952 odd numbers.
(3) Find the average of even numbers from 4 to 846
(4) Find the average of the first 4435 even numbers.
(5) What is the average of the first 1195 even numbers?
(6) What will be the average of the first 4330 odd numbers?
(7) Find the average of even numbers from 8 to 62
(8) Find the average of even numbers from 4 to 1640
(9) Find the average of the first 2568 odd numbers.
(10) Find the average of the first 2647 odd numbers.