Question : Find the average of even numbers from 4 to 1684
Correct Answer 844
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1684
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1684 are
4, 6, 8, . . . . 1684
After observing the above list of the even numbers from 4 to 1684 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1684 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1684
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1684
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 4 to 1684
= 4 + 1684/2
= 1688/2 = 844
Thus, the average of the even numbers from 4 to 1684 = 844 Answer
Method (2) to find the average of the even numbers from 4 to 1684
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1684 are
4, 6, 8, . . . . 1684
The even numbers from 4 to 1684 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1684
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 4 to 1684
1684 = 4 + (n – 1) × 2
⇒ 1684 = 4 + 2 n – 2
⇒ 1684 = 4 – 2 + 2 n
⇒ 1684 = 2 + 2 n
After transposing 2 to LHS
⇒ 1684 – 2 = 2 n
⇒ 1682 = 2 n
After rearranging the above expression
⇒ 2 n = 1682
After transposing 2 to RHS
⇒ n = 1682/2
⇒ n = 841
Thus, the number of terms of even numbers from 4 to 1684 = 841
This means 1684 is the 841th term.
Finding the sum of the given even numbers from 4 to 1684
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 4 to 1684
= 841/2 (4 + 1684)
= 841/2 × 1688
= 841 × 1688/2
= 1419608/2 = 709804
Thus, the sum of all terms of the given even numbers from 4 to 1684 = 709804
And, the total number of terms = 841
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 4 to 1684
= 709804/841 = 844
Thus, the average of the given even numbers from 4 to 1684 = 844 Answer
Similar Questions
(1) Find the average of the first 934 odd numbers.
(2) Find the average of the first 4126 even numbers.
(3) Find the average of the first 2645 odd numbers.
(4) Find the average of the first 3965 even numbers.
(5) Find the average of the first 3959 even numbers.
(6) What will be the average of the first 4965 odd numbers?
(7) Find the average of odd numbers from 9 to 17
(8) Find the average of the first 4714 even numbers.
(9) Find the average of odd numbers from 7 to 319
(10) What is the average of the first 23 even numbers?