Question:
Find the average of even numbers from 4 to 658
Correct Answer
331
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 658
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 658 are
4, 6, 8, . . . . 658
After observing the above list of the even numbers from 4 to 658 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 658 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 658
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 658
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 658
= 4 + 658/2
= 662/2 = 331
Thus, the average of the even numbers from 4 to 658 = 331 Answer
Method (2) to find the average of the even numbers from 4 to 658
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 658 are
4, 6, 8, . . . . 658
The even numbers from 4 to 658 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 658
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 658
658 = 4 + (n – 1) × 2
⇒ 658 = 4 + 2 n – 2
⇒ 658 = 4 – 2 + 2 n
⇒ 658 = 2 + 2 n
After transposing 2 to LHS
⇒ 658 – 2 = 2 n
⇒ 656 = 2 n
After rearranging the above expression
⇒ 2 n = 656
After transposing 2 to RHS
⇒ n = 656/2
⇒ n = 328
Thus, the number of terms of even numbers from 4 to 658 = 328
This means 658 is the 328th term.
Finding the sum of the given even numbers from 4 to 658
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 658
= 328/2 (4 + 658)
= 328/2 × 662
= 328 × 662/2
= 217136/2 = 108568
Thus, the sum of all terms of the given even numbers from 4 to 658 = 108568
And, the total number of terms = 328
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 658
= 108568/328 = 331
Thus, the average of the given even numbers from 4 to 658 = 331 Answer
Similar Questions
(1) Find the average of the first 3844 even numbers.
(2) Find the average of odd numbers from 5 to 545
(3) Find the average of the first 3089 even numbers.
(4) Find the average of the first 1748 odd numbers.
(5) Find the average of odd numbers from 13 to 701
(6) Find the average of odd numbers from 9 to 1335
(7) Find the average of the first 1983 odd numbers.
(8) Find the average of the first 4030 even numbers.
(9) What is the average of the first 1348 even numbers?
(10) Find the average of even numbers from 10 to 1086