Question : Find the average of even numbers from 4 to 530
Correct Answer 267
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 530
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 530 are
4, 6, 8, . . . . 530
After observing the above list of the even numbers from 4 to 530 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 530 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 530
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 530
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 530
= 4 + 530/2
= 534/2 = 267
Thus, the average of the even numbers from 4 to 530 = 267 Answer
Method (2) to find the average of the even numbers from 4 to 530
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 530 are
4, 6, 8, . . . . 530
The even numbers from 4 to 530 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 530
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 530
530 = 4 + (n – 1) × 2
⇒ 530 = 4 + 2 n – 2
⇒ 530 = 4 – 2 + 2 n
⇒ 530 = 2 + 2 n
After transposing 2 to LHS
⇒ 530 – 2 = 2 n
⇒ 528 = 2 n
After rearranging the above expression
⇒ 2 n = 528
After transposing 2 to RHS
⇒ n = 528/2
⇒ n = 264
Thus, the number of terms of even numbers from 4 to 530 = 264
This means 530 is the 264th term.
Finding the sum of the given even numbers from 4 to 530
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 530
= 264/2 (4 + 530)
= 264/2 × 534
= 264 × 534/2
= 140976/2 = 70488
Thus, the sum of all terms of the given even numbers from 4 to 530 = 70488
And, the total number of terms = 264
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 530
= 70488/264 = 267
Thus, the average of the given even numbers from 4 to 530 = 267 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 1043
(2) Find the average of odd numbers from 9 to 73
(3) Find the average of odd numbers from 7 to 35
(4) Find the average of odd numbers from 3 to 1101
(5) Find the average of odd numbers from 3 to 1253
(6) Find the average of odd numbers from 13 to 481
(7) Find the average of odd numbers from 5 to 891
(8) Find the average of even numbers from 12 to 34
(9) Find the average of even numbers from 12 to 124
(10) Find the average of even numbers from 4 to 116