Question:
Find the average of even numbers from 4 to 496
Correct Answer
250
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 496
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 496 are
4, 6, 8, . . . . 496
After observing the above list of the even numbers from 4 to 496 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 496 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 496
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 496
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 even numbers from 4 to 496
= 4 + 496/2
= 500/2 = 250
Thus, the average of the even numbers from 4 to 496 = 250 Answer
Method (2) to find the average of the even numbers from 4 to 496
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 496 are
4, 6, 8, . . . . 496
The even numbers from 4 to 496 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 496
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 even numbers from 4 to 496
496 = 4 + (n – 1) × 2
⇒ 496 = 4 + 2 n – 2
⇒ 496 = 4 – 2 + 2 n
⇒ 496 = 2 + 2 n
After transposing 2 to LHS
⇒ 496 – 2 = 2 n
⇒ 494 = 2 n
After rearranging the above expression
⇒ 2 n = 494
After transposing 2 to RHS
⇒ n = 494/2
⇒ n = 247
Thus, the number of terms of even numbers from 4 to 496 = 247
This means 496 is the 247th term.
Finding the sum of the given even numbers from 4 to 496
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 even numbers from 4 to 496
= 247/2 (4 + 496)
= 247/2 × 500
= 247 × 500/2
= 123500/2 = 61750
Thus, the sum of all terms of the given even numbers from 4 to 496 = 61750
And, the total number of terms = 247
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given even numbers from 4 to 496
= 61750/247 = 250
Thus, the average of the given even numbers from 4 to 496 = 250 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 1409
(2) Find the average of odd numbers from 15 to 1037
(3) What is the average of the first 1327 even numbers?
(4) Find the average of even numbers from 10 to 180
(5) Find the average of odd numbers from 15 to 1809
(6) Find the average of odd numbers from 13 to 1029
(7) What will be the average of the first 4284 odd numbers?
(8) Find the average of the first 2951 even numbers.
(9) What will be the average of the first 4768 odd numbers?
(10) What is the average of the first 1921 even numbers?