Question : Find the average of even numbers from 6 to 1682
Correct Answer 844
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1682
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1682 are
6, 8, 10, . . . . 1682
After observing the above list of the even numbers from 6 to 1682 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1682 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1682
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1682
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 1682
= 6 + 1682/2
= 1688/2 = 844
Thus, the average of the even numbers from 6 to 1682 = 844 Answer
Method (2) to find the average of the even numbers from 6 to 1682
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1682 are
6, 8, 10, . . . . 1682
The even numbers from 6 to 1682 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1682
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 1682
1682 = 6 + (n – 1) × 2
⇒ 1682 = 6 + 2 n – 2
⇒ 1682 = 6 – 2 + 2 n
⇒ 1682 = 4 + 2 n
After transposing 4 to LHS
⇒ 1682 – 4 = 2 n
⇒ 1678 = 2 n
After rearranging the above expression
⇒ 2 n = 1678
After transposing 2 to RHS
⇒ n = 1678/2
⇒ n = 839
Thus, the number of terms of even numbers from 6 to 1682 = 839
This means 1682 is the 839th term.
Finding the sum of the given even numbers from 6 to 1682
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 1682
= 839/2 (6 + 1682)
= 839/2 × 1688
= 839 × 1688/2
= 1416232/2 = 708116
Thus, the sum of all terms of the given even numbers from 6 to 1682 = 708116
And, the total number of terms = 839
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 1682
= 708116/839 = 844
Thus, the average of the given even numbers from 6 to 1682 = 844 Answer
Similar Questions
(1) Find the average of even numbers from 6 to 1844
(2) What will be the average of the first 4512 odd numbers?
(3) Find the average of even numbers from 4 to 710
(4) Find the average of odd numbers from 3 to 1357
(5) Find the average of the first 1882 odd numbers.
(6) Find the average of odd numbers from 9 to 387
(7) What is the average of the first 1911 even numbers?
(8) Find the average of odd numbers from 11 to 1387
(9) Find the average of even numbers from 10 to 1146
(10) Find the average of the first 1769 odd numbers.