Question:
Find the average of even numbers from 4 to 540
Correct Answer
272
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 540
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 540 are
4, 6, 8, . . . . 540
After observing the above list of the even numbers from 4 to 540 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 540 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 540
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 540
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 540
= 4 + 540/2
= 544/2 = 272
Thus, the average of the even numbers from 4 to 540 = 272 Answer
Method (2) to find the average of the even numbers from 4 to 540
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 540 are
4, 6, 8, . . . . 540
The even numbers from 4 to 540 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 540
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 540
540 = 4 + (n – 1) × 2
⇒ 540 = 4 + 2 n – 2
⇒ 540 = 4 – 2 + 2 n
⇒ 540 = 2 + 2 n
After transposing 2 to LHS
⇒ 540 – 2 = 2 n
⇒ 538 = 2 n
After rearranging the above expression
⇒ 2 n = 538
After transposing 2 to RHS
⇒ n = 538/2
⇒ n = 269
Thus, the number of terms of even numbers from 4 to 540 = 269
This means 540 is the 269th term.
Finding the sum of the given even numbers from 4 to 540
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 540
= 269/2 (4 + 540)
= 269/2 × 544
= 269 × 544/2
= 146336/2 = 73168
Thus, the sum of all terms of the given even numbers from 4 to 540 = 73168
And, the total number of terms = 269
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 540
= 73168/269 = 272
Thus, the average of the given even numbers from 4 to 540 = 272 Answer
Similar Questions
(1) Find the average of the first 2725 odd numbers.
(2) What will be the average of the first 4147 odd numbers?
(3) What is the average of the first 1449 even numbers?
(4) Find the average of odd numbers from 7 to 1159
(5) What will be the average of the first 4899 odd numbers?
(6) Find the average of even numbers from 12 to 654
(7) Find the average of even numbers from 4 to 1808
(8) Find the average of the first 2814 odd numbers.
(9) Find the average of odd numbers from 15 to 1095
(10) What is the average of the first 871 even numbers?