Question : Find the average of even numbers from 6 to 634
Correct Answer 320
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 634
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 634 are
6, 8, 10, . . . . 634
After observing the above list of the even numbers from 6 to 634 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 634 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 634
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 634
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 634
= 6 + 634/2
= 640/2 = 320
Thus, the average of the even numbers from 6 to 634 = 320 Answer
Method (2) to find the average of the even numbers from 6 to 634
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 634 are
6, 8, 10, . . . . 634
The even numbers from 6 to 634 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 634
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 634
634 = 6 + (n – 1) × 2
⇒ 634 = 6 + 2 n – 2
⇒ 634 = 6 – 2 + 2 n
⇒ 634 = 4 + 2 n
After transposing 4 to LHS
⇒ 634 – 4 = 2 n
⇒ 630 = 2 n
After rearranging the above expression
⇒ 2 n = 630
After transposing 2 to RHS
⇒ n = 630/2
⇒ n = 315
Thus, the number of terms of even numbers from 6 to 634 = 315
This means 634 is the 315th term.
Finding the sum of the given even numbers from 6 to 634
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 634
= 315/2 (6 + 634)
= 315/2 × 640
= 315 × 640/2
= 201600/2 = 100800
Thus, the sum of all terms of the given even numbers from 6 to 634 = 100800
And, the total number of terms = 315
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 634
= 100800/315 = 320
Thus, the average of the given even numbers from 6 to 634 = 320 Answer
Similar Questions
(1) What is the average of the first 689 even numbers?
(2) Find the average of even numbers from 10 to 1594
(3) Find the average of even numbers from 10 to 204
(4) What is the average of the first 297 even numbers?
(5) Find the average of the first 1203 odd numbers.
(6) Find the average of even numbers from 8 to 1100
(7) Find the average of even numbers from 6 to 380
(8) What is the average of the first 58 odd numbers?
(9) Find the average of odd numbers from 5 to 519
(10) Find the average of even numbers from 10 to 360