Question:
Find the average of even numbers from 6 to 1568
Correct Answer
787
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1568
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1568 are
6, 8, 10, . . . . 1568
After observing the above list of the even numbers from 6 to 1568 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1568 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1568
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1568
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 1568
= 6 + 1568/2
= 1574/2 = 787
Thus, the average of the even numbers from 6 to 1568 = 787 Answer
Method (2) to find the average of the even numbers from 6 to 1568
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1568 are
6, 8, 10, . . . . 1568
The even numbers from 6 to 1568 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1568
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 1568
1568 = 6 + (n – 1) × 2
⇒ 1568 = 6 + 2 n – 2
⇒ 1568 = 6 – 2 + 2 n
⇒ 1568 = 4 + 2 n
After transposing 4 to LHS
⇒ 1568 – 4 = 2 n
⇒ 1564 = 2 n
After rearranging the above expression
⇒ 2 n = 1564
After transposing 2 to RHS
⇒ n = 1564/2
⇒ n = 782
Thus, the number of terms of even numbers from 6 to 1568 = 782
This means 1568 is the 782th term.
Finding the sum of the given even numbers from 6 to 1568
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 1568
= 782/2 (6 + 1568)
= 782/2 × 1574
= 782 × 1574/2
= 1230868/2 = 615434
Thus, the sum of all terms of the given even numbers from 6 to 1568 = 615434
And, the total number of terms = 782
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 1568
= 615434/782 = 787
Thus, the average of the given even numbers from 6 to 1568 = 787 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 435
(2) Find the average of odd numbers from 15 to 817
(3) Find the average of the first 2317 odd numbers.
(4) Find the average of the first 2176 even numbers.
(5) Find the average of the first 2607 even numbers.
(6) What is the average of the first 78 even numbers?
(7) Find the average of the first 4805 even numbers.
(8) Find the average of even numbers from 12 to 1006
(9) Find the average of the first 4629 even numbers.
(10) What is the average of the first 1451 even numbers?