Question : Find the average of odd numbers from 13 to 457
Correct Answer 235
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 457
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 457 are
13, 15, 17, . . . . 457
After observing the above list of the odd numbers from 13 to 457 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 457 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 457
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 457
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 457
= 13 + 457/2
= 470/2 = 235
Thus, the average of the odd numbers from 13 to 457 = 235 Answer
Method (2) to find the average of the odd numbers from 13 to 457
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 457 are
13, 15, 17, . . . . 457
The odd numbers from 13 to 457 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 457
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 457
457 = 13 + (n – 1) × 2
⇒ 457 = 13 + 2 n – 2
⇒ 457 = 13 – 2 + 2 n
⇒ 457 = 11 + 2 n
After transposing 11 to LHS
⇒ 457 – 11 = 2 n
⇒ 446 = 2 n
After rearranging the above expression
⇒ 2 n = 446
After transposing 2 to RHS
⇒ n = 446/2
⇒ n = 223
Thus, the number of terms of odd numbers from 13 to 457 = 223
This means 457 is the 223th term.
Finding the sum of the given odd numbers from 13 to 457
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 457
= 223/2 (13 + 457)
= 223/2 × 470
= 223 × 470/2
= 104810/2 = 52405
Thus, the sum of all terms of the given odd numbers from 13 to 457 = 52405
And, the total number of terms = 223
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 457
= 52405/223 = 235
Thus, the average of the given odd numbers from 13 to 457 = 235 Answer
Similar Questions
(1) Find the average of the first 3051 even numbers.
(2) Find the average of the first 391 odd numbers.
(3) What will be the average of the first 4619 odd numbers?
(4) Find the average of odd numbers from 15 to 23
(5) Find the average of the first 1136 odd numbers.
(6) Find the average of even numbers from 8 to 680
(7) Find the average of the first 2077 even numbers.
(8) What is the average of the first 773 even numbers?
(9) What will be the average of the first 4487 odd numbers?
(10) Find the average of even numbers from 4 to 1950