Question:
Find the average of even numbers from 4 to 996
Correct Answer
500
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 996
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 996 are
4, 6, 8, . . . . 996
After observing the above list of the even numbers from 4 to 996 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 996 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 996
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 996
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 996
= 4 + 996/2
= 1000/2 = 500
Thus, the average of the even numbers from 4 to 996 = 500 Answer
Method (2) to find the average of the even numbers from 4 to 996
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 996 are
4, 6, 8, . . . . 996
The even numbers from 4 to 996 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 996
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 996
996 = 4 + (n – 1) × 2
⇒ 996 = 4 + 2 n – 2
⇒ 996 = 4 – 2 + 2 n
⇒ 996 = 2 + 2 n
After transposing 2 to LHS
⇒ 996 – 2 = 2 n
⇒ 994 = 2 n
After rearranging the above expression
⇒ 2 n = 994
After transposing 2 to RHS
⇒ n = 994/2
⇒ n = 497
Thus, the number of terms of even numbers from 4 to 996 = 497
This means 996 is the 497th term.
Finding the sum of the given even numbers from 4 to 996
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 996
= 497/2 (4 + 996)
= 497/2 × 1000
= 497 × 1000/2
= 497000/2 = 248500
Thus, the sum of all terms of the given even numbers from 4 to 996 = 248500
And, the total number of terms = 497
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 996
= 248500/497 = 500
Thus, the average of the given even numbers from 4 to 996 = 500 Answer
Similar Questions
(1) Find the average of the first 2291 even numbers.
(2) Find the average of the first 3902 odd numbers.
(3) Find the average of odd numbers from 5 to 1261
(4) Find the average of odd numbers from 9 to 1389
(5) Find the average of the first 3052 even numbers.
(6) Find the average of the first 4892 even numbers.
(7) Find the average of odd numbers from 11 to 237
(8) Find the average of the first 2296 odd numbers.
(9) What will be the average of the first 4298 odd numbers?
(10) Find the average of even numbers from 4 to 1676