Question:
Find the average of odd numbers from 13 to 23
Correct Answer
18
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 23
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 23 are
13, 15, 17, . . . . 23
After observing the above list of the odd numbers from 13 to 23 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 23 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 23
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 23
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 23
= 13 + 23/2
= 36/2 = 18
Thus, the average of the odd numbers from 13 to 23 = 18 Answer
Method (2) to find the average of the odd numbers from 13 to 23
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 23 are
13, 15, 17, . . . . 23
The odd numbers from 13 to 23 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 23
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 23
23 = 13 + (n – 1) × 2
⇒ 23 = 13 + 2 n – 2
⇒ 23 = 13 – 2 + 2 n
⇒ 23 = 11 + 2 n
After transposing 11 to LHS
⇒ 23 – 11 = 2 n
⇒ 12 = 2 n
After rearranging the above expression
⇒ 2 n = 12
After transposing 2 to RHS
⇒ n = 12/2
⇒ n = 6
Thus, the number of terms of odd numbers from 13 to 23 = 6
This means 23 is the 6th term.
Finding the sum of the given odd numbers from 13 to 23
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 23
= 6/2 (13 + 23)
= 6/2 × 36
= 6 × 36/2
= 216/2 = 108
Thus, the sum of all terms of the given odd numbers from 13 to 23 = 108
And, the total number of terms = 6
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 23
= 108/6 = 18
Thus, the average of the given odd numbers from 13 to 23 = 18 Answer
Similar Questions
(1) Find the average of the first 1728 odd numbers.
(2) Find the average of even numbers from 10 to 118
(3) Find the average of the first 3210 even numbers.
(4) What is the average of the first 840 even numbers?
(5) Find the average of even numbers from 6 to 984
(6) Find the average of even numbers from 8 to 384
(7) Find the average of the first 3150 odd numbers.
(8) Find the average of even numbers from 10 to 462
(9) Find the average of even numbers from 4 to 436
(10) Find the average of odd numbers from 15 to 919