Question : Find the average of even numbers from 4 to 878
Correct Answer 441
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 878
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 878 are
4, 6, 8, . . . . 878
After observing the above list of the even numbers from 4 to 878 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 878 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 878
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 878
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 878
= 4 + 878/2
= 882/2 = 441
Thus, the average of the even numbers from 4 to 878 = 441 Answer
Method (2) to find the average of the even numbers from 4 to 878
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 878 are
4, 6, 8, . . . . 878
The even numbers from 4 to 878 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 878
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 878
878 = 4 + (n – 1) × 2
⇒ 878 = 4 + 2 n – 2
⇒ 878 = 4 – 2 + 2 n
⇒ 878 = 2 + 2 n
After transposing 2 to LHS
⇒ 878 – 2 = 2 n
⇒ 876 = 2 n
After rearranging the above expression
⇒ 2 n = 876
After transposing 2 to RHS
⇒ n = 876/2
⇒ n = 438
Thus, the number of terms of even numbers from 4 to 878 = 438
This means 878 is the 438th term.
Finding the sum of the given even numbers from 4 to 878
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 878
= 438/2 (4 + 878)
= 438/2 × 882
= 438 × 882/2
= 386316/2 = 193158
Thus, the sum of all terms of the given even numbers from 4 to 878 = 193158
And, the total number of terms = 438
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 878
= 193158/438 = 441
Thus, the average of the given even numbers from 4 to 878 = 441 Answer
Similar Questions
(1) Find the average of the first 3153 odd numbers.
(2) Find the average of even numbers from 4 to 1436
(3) Find the average of the first 2646 odd numbers.
(4) Find the average of the first 2033 odd numbers.
(5) Find the average of the first 2233 even numbers.
(6) Find the average of even numbers from 10 to 1224
(7) Find the average of even numbers from 4 to 1964
(8) Find the average of even numbers from 6 to 1170
(9) Find the average of even numbers from 10 to 1684
(10) Find the average of the first 1700 odd numbers.