Question:
Find the average of even numbers from 4 to 484
Correct Answer
244
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 484
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 484 are
4, 6, 8, . . . . 484
After observing the above list of the even numbers from 4 to 484 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 484 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 484
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 484
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 484
= 4 + 484/2
= 488/2 = 244
Thus, the average of the even numbers from 4 to 484 = 244 Answer
Method (2) to find the average of the even numbers from 4 to 484
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 484 are
4, 6, 8, . . . . 484
The even numbers from 4 to 484 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 484
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 484
484 = 4 + (n – 1) × 2
⇒ 484 = 4 + 2 n – 2
⇒ 484 = 4 – 2 + 2 n
⇒ 484 = 2 + 2 n
After transposing 2 to LHS
⇒ 484 – 2 = 2 n
⇒ 482 = 2 n
After rearranging the above expression
⇒ 2 n = 482
After transposing 2 to RHS
⇒ n = 482/2
⇒ n = 241
Thus, the number of terms of even numbers from 4 to 484 = 241
This means 484 is the 241th term.
Finding the sum of the given even numbers from 4 to 484
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 484
= 241/2 (4 + 484)
= 241/2 × 488
= 241 × 488/2
= 117608/2 = 58804
Thus, the sum of all terms of the given even numbers from 4 to 484 = 58804
And, the total number of terms = 241
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 484
= 58804/241 = 244
Thus, the average of the given even numbers from 4 to 484 = 244 Answer
Similar Questions
(1) Find the average of the first 2732 even numbers.
(2) Find the average of odd numbers from 5 to 713
(3) Find the average of the first 3107 odd numbers.
(4) Find the average of odd numbers from 13 to 677
(5) Find the average of even numbers from 4 to 1066
(6) Find the average of odd numbers from 13 to 1021
(7) What is the average of the first 1979 even numbers?
(8) What is the average of the first 942 even numbers?
(9) Find the average of the first 1011 odd numbers.
(10) Find the average of the first 4913 even numbers.