Question : Find the average of even numbers from 4 to 1492
Correct Answer 748
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1492
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1492 are
4, 6, 8, . . . . 1492
After observing the above list of the even numbers from 4 to 1492 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1492 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1492
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1492
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 1492
= 4 + 1492/2
= 1496/2 = 748
Thus, the average of the even numbers from 4 to 1492 = 748 Answer
Method (2) to find the average of the even numbers from 4 to 1492
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1492 are
4, 6, 8, . . . . 1492
The even numbers from 4 to 1492 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1492
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 1492
1492 = 4 + (n – 1) × 2
⇒ 1492 = 4 + 2 n – 2
⇒ 1492 = 4 – 2 + 2 n
⇒ 1492 = 2 + 2 n
After transposing 2 to LHS
⇒ 1492 – 2 = 2 n
⇒ 1490 = 2 n
After rearranging the above expression
⇒ 2 n = 1490
After transposing 2 to RHS
⇒ n = 1490/2
⇒ n = 745
Thus, the number of terms of even numbers from 4 to 1492 = 745
This means 1492 is the 745th term.
Finding the sum of the given even numbers from 4 to 1492
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 1492
= 745/2 (4 + 1492)
= 745/2 × 1496
= 745 × 1496/2
= 1114520/2 = 557260
Thus, the sum of all terms of the given even numbers from 4 to 1492 = 557260
And, the total number of terms = 745
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 1492
= 557260/745 = 748
Thus, the average of the given even numbers from 4 to 1492 = 748 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 400
(2) What is the average of the first 40 even numbers?
(3) Find the average of even numbers from 4 to 1030
(4) Find the average of the first 3933 odd numbers.
(5) Find the average of odd numbers from 13 to 1361
(6) Find the average of even numbers from 6 to 930
(7) Find the average of the first 1205 odd numbers.
(8) What is the average of the first 1887 even numbers?
(9) What will be the average of the first 4877 odd numbers?
(10) Find the average of the first 2528 even numbers.