Question:
Find the average of even numbers from 8 to 828
Correct Answer
418
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 828
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 828 are
8, 10, 12, . . . . 828
After observing the above list of the even numbers from 8 to 828 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 828 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 828
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 828
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 8 to 828
= 8 + 828/2
= 836/2 = 418
Thus, the average of the even numbers from 8 to 828 = 418 Answer
Method (2) to find the average of the even numbers from 8 to 828
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 828 are
8, 10, 12, . . . . 828
The even numbers from 8 to 828 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 828
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 8 to 828
828 = 8 + (n – 1) × 2
⇒ 828 = 8 + 2 n – 2
⇒ 828 = 8 – 2 + 2 n
⇒ 828 = 6 + 2 n
After transposing 6 to LHS
⇒ 828 – 6 = 2 n
⇒ 822 = 2 n
After rearranging the above expression
⇒ 2 n = 822
After transposing 2 to RHS
⇒ n = 822/2
⇒ n = 411
Thus, the number of terms of even numbers from 8 to 828 = 411
This means 828 is the 411th term.
Finding the sum of the given even numbers from 8 to 828
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 8 to 828
= 411/2 (8 + 828)
= 411/2 × 836
= 411 × 836/2
= 343596/2 = 171798
Thus, the sum of all terms of the given even numbers from 8 to 828 = 171798
And, the total number of terms = 411
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 8 to 828
= 171798/411 = 418
Thus, the average of the given even numbers from 8 to 828 = 418 Answer
Similar Questions
(1) Find the average of even numbers from 4 to 1292
(2) Find the average of even numbers from 12 to 1278
(3) What is the average of the first 1938 even numbers?
(4) Find the average of the first 2152 even numbers.
(5) Find the average of odd numbers from 13 to 485
(6) Find the average of even numbers from 12 to 1798
(7) Find the average of even numbers from 12 to 650
(8) Find the average of the first 2688 odd numbers.
(9) Find the average of the first 4469 even numbers.
(10) Find the average of the first 1975 odd numbers.