Question:
Find the average of even numbers from 4 to 1728
Correct Answer
866
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1728
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1728 are
4, 6, 8, . . . . 1728
After observing the above list of the even numbers from 4 to 1728 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1728 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1728
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1728
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 1728
= 4 + 1728/2
= 1732/2 = 866
Thus, the average of the even numbers from 4 to 1728 = 866 Answer
Method (2) to find the average of the even numbers from 4 to 1728
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1728 are
4, 6, 8, . . . . 1728
The even numbers from 4 to 1728 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1728
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 1728
1728 = 4 + (n – 1) × 2
⇒ 1728 = 4 + 2 n – 2
⇒ 1728 = 4 – 2 + 2 n
⇒ 1728 = 2 + 2 n
After transposing 2 to LHS
⇒ 1728 – 2 = 2 n
⇒ 1726 = 2 n
After rearranging the above expression
⇒ 2 n = 1726
After transposing 2 to RHS
⇒ n = 1726/2
⇒ n = 863
Thus, the number of terms of even numbers from 4 to 1728 = 863
This means 1728 is the 863th term.
Finding the sum of the given even numbers from 4 to 1728
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 1728
= 863/2 (4 + 1728)
= 863/2 × 1732
= 863 × 1732/2
= 1494716/2 = 747358
Thus, the sum of all terms of the given even numbers from 4 to 1728 = 747358
And, the total number of terms = 863
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 1728
= 747358/863 = 866
Thus, the average of the given even numbers from 4 to 1728 = 866 Answer
Similar Questions
(1) What is the average of the first 1662 even numbers?
(2) Find the average of odd numbers from 15 to 257
(3) Find the average of even numbers from 10 to 334
(4) Find the average of even numbers from 12 to 1018
(5) Find the average of odd numbers from 15 to 381
(6) Find the average of odd numbers from 13 to 83
(7) Find the average of odd numbers from 13 to 1109
(8) Find the average of the first 1506 odd numbers.
(9) Find the average of odd numbers from 9 to 465
(10) Find the average of the first 3872 even numbers.