Question:
Find the average of even numbers from 6 to 1656
Correct Answer
831
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1656
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1656 are
6, 8, 10, . . . . 1656
After observing the above list of the even numbers from 6 to 1656 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1656 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1656
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1656
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 1656
= 6 + 1656/2
= 1662/2 = 831
Thus, the average of the even numbers from 6 to 1656 = 831 Answer
Method (2) to find the average of the even numbers from 6 to 1656
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1656 are
6, 8, 10, . . . . 1656
The even numbers from 6 to 1656 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1656
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 1656
1656 = 6 + (n – 1) × 2
⇒ 1656 = 6 + 2 n – 2
⇒ 1656 = 6 – 2 + 2 n
⇒ 1656 = 4 + 2 n
After transposing 4 to LHS
⇒ 1656 – 4 = 2 n
⇒ 1652 = 2 n
After rearranging the above expression
⇒ 2 n = 1652
After transposing 2 to RHS
⇒ n = 1652/2
⇒ n = 826
Thus, the number of terms of even numbers from 6 to 1656 = 826
This means 1656 is the 826th term.
Finding the sum of the given even numbers from 6 to 1656
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 1656
= 826/2 (6 + 1656)
= 826/2 × 1662
= 826 × 1662/2
= 1372812/2 = 686406
Thus, the sum of all terms of the given even numbers from 6 to 1656 = 686406
And, the total number of terms = 826
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 1656
= 686406/826 = 831
Thus, the average of the given even numbers from 6 to 1656 = 831 Answer
Similar Questions
(1) Find the average of even numbers from 6 to 1008
(2) Find the average of even numbers from 8 to 836
(3) What is the average of the first 1682 even numbers?
(4) Find the average of the first 589 odd numbers.
(5) Find the average of the first 2098 odd numbers.
(6) Find the average of even numbers from 12 to 106
(7) Find the average of even numbers from 12 to 438
(8) Find the average of the first 3139 even numbers.
(9) Find the average of odd numbers from 11 to 131
(10) Find the average of the first 2434 odd numbers.