Question:
Find the average of even numbers from 4 to 1820
Correct Answer
912
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1820
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1820 are
4, 6, 8, . . . . 1820
After observing the above list of the even numbers from 4 to 1820 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1820 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1820
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1820
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 1820
= 4 + 1820/2
= 1824/2 = 912
Thus, the average of the even numbers from 4 to 1820 = 912 Answer
Method (2) to find the average of the even numbers from 4 to 1820
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1820 are
4, 6, 8, . . . . 1820
The even numbers from 4 to 1820 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1820
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 1820
1820 = 4 + (n – 1) × 2
⇒ 1820 = 4 + 2 n – 2
⇒ 1820 = 4 – 2 + 2 n
⇒ 1820 = 2 + 2 n
After transposing 2 to LHS
⇒ 1820 – 2 = 2 n
⇒ 1818 = 2 n
After rearranging the above expression
⇒ 2 n = 1818
After transposing 2 to RHS
⇒ n = 1818/2
⇒ n = 909
Thus, the number of terms of even numbers from 4 to 1820 = 909
This means 1820 is the 909th term.
Finding the sum of the given even numbers from 4 to 1820
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 1820
= 909/2 (4 + 1820)
= 909/2 × 1824
= 909 × 1824/2
= 1658016/2 = 829008
Thus, the sum of all terms of the given even numbers from 4 to 1820 = 829008
And, the total number of terms = 909
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 1820
= 829008/909 = 912
Thus, the average of the given even numbers from 4 to 1820 = 912 Answer
Similar Questions
(1) What is the average of the first 768 even numbers?
(2) Find the average of odd numbers from 15 to 333
(3) Find the average of the first 3943 even numbers.
(4) What is the average of the first 900 even numbers?
(5) Find the average of odd numbers from 3 to 1073
(6) Find the average of odd numbers from 3 to 273
(7) Find the average of even numbers from 12 to 1658
(8) Find the average of odd numbers from 7 to 1195
(9) Find the average of the first 3155 odd numbers.
(10) Find the average of odd numbers from 5 to 567