Question:
Find the average of even numbers from 6 to 1696
Correct Answer
851
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1696
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1696 are
6, 8, 10, . . . . 1696
After observing the above list of the even numbers from 6 to 1696 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1696 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1696
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1696
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 1696
= 6 + 1696/2
= 1702/2 = 851
Thus, the average of the even numbers from 6 to 1696 = 851 Answer
Method (2) to find the average of the even numbers from 6 to 1696
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1696 are
6, 8, 10, . . . . 1696
The even numbers from 6 to 1696 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1696
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 1696
1696 = 6 + (n – 1) × 2
⇒ 1696 = 6 + 2 n – 2
⇒ 1696 = 6 – 2 + 2 n
⇒ 1696 = 4 + 2 n
After transposing 4 to LHS
⇒ 1696 – 4 = 2 n
⇒ 1692 = 2 n
After rearranging the above expression
⇒ 2 n = 1692
After transposing 2 to RHS
⇒ n = 1692/2
⇒ n = 846
Thus, the number of terms of even numbers from 6 to 1696 = 846
This means 1696 is the 846th term.
Finding the sum of the given even numbers from 6 to 1696
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 1696
= 846/2 (6 + 1696)
= 846/2 × 1702
= 846 × 1702/2
= 1439892/2 = 719946
Thus, the sum of all terms of the given even numbers from 6 to 1696 = 719946
And, the total number of terms = 846
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 1696
= 719946/846 = 851
Thus, the average of the given even numbers from 6 to 1696 = 851 Answer
Similar Questions
(1) Find the average of the first 4005 even numbers.
(2) Find the average of even numbers from 4 to 1920
(3) Find the average of the first 3664 even numbers.
(4) Find the average of odd numbers from 7 to 621
(5) Find the average of the first 4064 even numbers.
(6) Find the average of the first 1739 odd numbers.
(7) Find the average of even numbers from 6 to 674
(8) Find the average of even numbers from 12 to 938
(9) Find the average of the first 1958 odd numbers.
(10) Find the average of the first 4751 even numbers.