Question : Find the average of odd numbers from 13 to 623
Correct Answer 318
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 623
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 623 are
13, 15, 17, . . . . 623
After observing the above list of the odd numbers from 13 to 623 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 623 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 623
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 623
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 623
= 13 + 623/2
= 636/2 = 318
Thus, the average of the odd numbers from 13 to 623 = 318 Answer
Method (2) to find the average of the odd numbers from 13 to 623
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 623 are
13, 15, 17, . . . . 623
The odd numbers from 13 to 623 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 623
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 623
623 = 13 + (n – 1) × 2
⇒ 623 = 13 + 2 n – 2
⇒ 623 = 13 – 2 + 2 n
⇒ 623 = 11 + 2 n
After transposing 11 to LHS
⇒ 623 – 11 = 2 n
⇒ 612 = 2 n
After rearranging the above expression
⇒ 2 n = 612
After transposing 2 to RHS
⇒ n = 612/2
⇒ n = 306
Thus, the number of terms of odd numbers from 13 to 623 = 306
This means 623 is the 306th term.
Finding the sum of the given odd numbers from 13 to 623
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 623
= 306/2 (13 + 623)
= 306/2 × 636
= 306 × 636/2
= 194616/2 = 97308
Thus, the sum of all terms of the given odd numbers from 13 to 623 = 97308
And, the total number of terms = 306
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 623
= 97308/306 = 318
Thus, the average of the given odd numbers from 13 to 623 = 318 Answer
Similar Questions
(1) Find the average of the first 2968 odd numbers.
(2) Find the average of odd numbers from 3 to 95
(3) Find the average of even numbers from 12 to 662
(4) Find the average of the first 3898 odd numbers.
(5) Find the average of odd numbers from 11 to 1473
(6) Find the average of the first 2681 odd numbers.
(7) Find the average of the first 2531 odd numbers.
(8) Find the average of the first 2240 even numbers.
(9) What is the average of the first 886 even numbers?
(10) Find the average of even numbers from 10 to 438