Question:
Find the average of even numbers from 6 to 590
Correct Answer
298
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 590
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 590 are
6, 8, 10, . . . . 590
After observing the above list of the even numbers from 6 to 590 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 590 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 590
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 590
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 590
= 6 + 590/2
= 596/2 = 298
Thus, the average of the even numbers from 6 to 590 = 298 Answer
Method (2) to find the average of the even numbers from 6 to 590
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 590 are
6, 8, 10, . . . . 590
The even numbers from 6 to 590 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 590
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 590
590 = 6 + (n – 1) × 2
⇒ 590 = 6 + 2 n – 2
⇒ 590 = 6 – 2 + 2 n
⇒ 590 = 4 + 2 n
After transposing 4 to LHS
⇒ 590 – 4 = 2 n
⇒ 586 = 2 n
After rearranging the above expression
⇒ 2 n = 586
After transposing 2 to RHS
⇒ n = 586/2
⇒ n = 293
Thus, the number of terms of even numbers from 6 to 590 = 293
This means 590 is the 293th term.
Finding the sum of the given even numbers from 6 to 590
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 590
= 293/2 (6 + 590)
= 293/2 × 596
= 293 × 596/2
= 174628/2 = 87314
Thus, the sum of all terms of the given even numbers from 6 to 590 = 87314
And, the total number of terms = 293
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 590
= 87314/293 = 298
Thus, the average of the given even numbers from 6 to 590 = 298 Answer
Similar Questions
(1) Find the average of the first 2598 even numbers.
(2) Find the average of the first 4133 even numbers.
(3) Find the average of even numbers from 6 to 1394
(4) Find the average of even numbers from 4 to 618
(5) Find the average of the first 2406 even numbers.
(6) Find the average of odd numbers from 5 to 1141
(7) Find the average of the first 2437 odd numbers.
(8) Find the average of the first 3711 odd numbers.
(9) What will be the average of the first 4353 odd numbers?
(10) Find the average of odd numbers from 11 to 695