Question:
( 1 of 10 ) Find the average of even numbers from 4 to 1870
(A) 24
(B) 25
(C) 36
(D) 23
You selected
938
Correct Answer
937
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1870
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1870 are
4, 6, 8, . . . . 1870
After observing the above list of the even numbers from 4 to 1870 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1870 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1870
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1870
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 1870
= 4 + 1870/2
= 1874/2 = 937
Thus, the average of the even numbers from 4 to 1870 = 937 Answer
Method (2) to find the average of the even numbers from 4 to 1870
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1870 are
4, 6, 8, . . . . 1870
The even numbers from 4 to 1870 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1870
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 1870
1870 = 4 + (n – 1) × 2
⇒ 1870 = 4 + 2 n – 2
⇒ 1870 = 4 – 2 + 2 n
⇒ 1870 = 2 + 2 n
After transposing 2 to LHS
⇒ 1870 – 2 = 2 n
⇒ 1868 = 2 n
After rearranging the above expression
⇒ 2 n = 1868
After transposing 2 to RHS
⇒ n = 1868/2
⇒ n = 934
Thus, the number of terms of even numbers from 4 to 1870 = 934
This means 1870 is the 934th term.
Finding the sum of the given even numbers from 4 to 1870
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 1870
= 934/2 (4 + 1870)
= 934/2 × 1874
= 934 × 1874/2
= 1750316/2 = 875158
Thus, the sum of all terms of the given even numbers from 4 to 1870 = 875158
And, the total number of terms = 934
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 1870
= 875158/934 = 937
Thus, the average of the given even numbers from 4 to 1870 = 937 Answer
Similar Questions
(1) What will be the average of the first 4781 odd numbers?
(2) What is the average of the first 1574 even numbers?
(3) Find the average of odd numbers from 9 to 401
(4) Find the average of odd numbers from 3 to 1091
(5) Find the average of the first 3101 odd numbers.
(6) Find the average of the first 3798 odd numbers.
(7) Find the average of even numbers from 12 to 1892
(8) Find the average of odd numbers from 11 to 1145
(9) Find the average of odd numbers from 7 to 123
(10) What is the average of the first 1391 even numbers?