Question : Find the average of even numbers from 6 to 838
Correct Answer 422
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 838
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 838 are
6, 8, 10, . . . . 838
After observing the above list of the even numbers from 6 to 838 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 838 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 838
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 838
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 6 to 838
= 6 + 838/2
= 844/2 = 422
Thus, the average of the even numbers from 6 to 838 = 422 Answer
Method (2) to find the average of the even numbers from 6 to 838
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 838 are
6, 8, 10, . . . . 838
The even numbers from 6 to 838 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 838
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 6 to 838
838 = 6 + (n – 1) × 2
⇒ 838 = 6 + 2 n – 2
⇒ 838 = 6 – 2 + 2 n
⇒ 838 = 4 + 2 n
After transposing 4 to LHS
⇒ 838 – 4 = 2 n
⇒ 834 = 2 n
After rearranging the above expression
⇒ 2 n = 834
After transposing 2 to RHS
⇒ n = 834/2
⇒ n = 417
Thus, the number of terms of even numbers from 6 to 838 = 417
This means 838 is the 417th term.
Finding the sum of the given even numbers from 6 to 838
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 6 to 838
= 417/2 (6 + 838)
= 417/2 × 844
= 417 × 844/2
= 351948/2 = 175974
Thus, the sum of all terms of the given even numbers from 6 to 838 = 175974
And, the total number of terms = 417
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 6 to 838
= 175974/417 = 422
Thus, the average of the given even numbers from 6 to 838 = 422 Answer
Similar Questions
(1) Find the average of the first 881 odd numbers.
(2) Find the average of even numbers from 10 to 1704
(3) Find the average of the first 665 odd numbers.
(4) Find the average of even numbers from 8 to 26
(5) Find the average of odd numbers from 13 to 851
(6) Find the average of the first 3613 odd numbers.
(7) Find the average of the first 3239 odd numbers.
(8) Find the average of odd numbers from 15 to 415
(9) Find the average of the first 1123 odd numbers.
(10) What will be the average of the first 4798 odd numbers?