Question:
Find the average of even numbers from 4 to 444
Correct Answer
224
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 444
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 444 are
4, 6, 8, . . . . 444
After observing the above list of the even numbers from 4 to 444 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 444 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 444
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 444
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 444
= 4 + 444/2
= 448/2 = 224
Thus, the average of the even numbers from 4 to 444 = 224 Answer
Method (2) to find the average of the even numbers from 4 to 444
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 444 are
4, 6, 8, . . . . 444
The even numbers from 4 to 444 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 444
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 444
444 = 4 + (n – 1) × 2
⇒ 444 = 4 + 2 n – 2
⇒ 444 = 4 – 2 + 2 n
⇒ 444 = 2 + 2 n
After transposing 2 to LHS
⇒ 444 – 2 = 2 n
⇒ 442 = 2 n
After rearranging the above expression
⇒ 2 n = 442
After transposing 2 to RHS
⇒ n = 442/2
⇒ n = 221
Thus, the number of terms of even numbers from 4 to 444 = 221
This means 444 is the 221th term.
Finding the sum of the given even numbers from 4 to 444
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 444
= 221/2 (4 + 444)
= 221/2 × 448
= 221 × 448/2
= 99008/2 = 49504
Thus, the sum of all terms of the given even numbers from 4 to 444 = 49504
And, the total number of terms = 221
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 444
= 49504/221 = 224
Thus, the average of the given even numbers from 4 to 444 = 224 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 95
(2) Find the average of odd numbers from 13 to 163
(3) What is the average of the first 1139 even numbers?
(4) What will be the average of the first 4529 odd numbers?
(5) Find the average of odd numbers from 15 to 1757
(6) Find the average of odd numbers from 5 to 185
(7) Find the average of the first 2969 even numbers.
(8) Find the average of odd numbers from 11 to 615
(9) Find the average of the first 4288 even numbers.
(10) Find the average of odd numbers from 11 to 1183