Question : Find the average of even numbers from 12 to 1824
Correct Answer 918
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1824
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1824 are
12, 14, 16, . . . . 1824
After observing the above list of the even numbers from 12 to 1824 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1824 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1824
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1824
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 12 to 1824
= 12 + 1824/2
= 1836/2 = 918
Thus, the average of the even numbers from 12 to 1824 = 918 Answer
Method (2) to find the average of the even numbers from 12 to 1824
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1824 are
12, 14, 16, . . . . 1824
The even numbers from 12 to 1824 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1824
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 12 to 1824
1824 = 12 + (n – 1) × 2
⇒ 1824 = 12 + 2 n – 2
⇒ 1824 = 12 – 2 + 2 n
⇒ 1824 = 10 + 2 n
After transposing 10 to LHS
⇒ 1824 – 10 = 2 n
⇒ 1814 = 2 n
After rearranging the above expression
⇒ 2 n = 1814
After transposing 2 to RHS
⇒ n = 1814/2
⇒ n = 907
Thus, the number of terms of even numbers from 12 to 1824 = 907
This means 1824 is the 907th term.
Finding the sum of the given even numbers from 12 to 1824
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 12 to 1824
= 907/2 (12 + 1824)
= 907/2 × 1836
= 907 × 1836/2
= 1665252/2 = 832626
Thus, the sum of all terms of the given even numbers from 12 to 1824 = 832626
And, the total number of terms = 907
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 12 to 1824
= 832626/907 = 918
Thus, the average of the given even numbers from 12 to 1824 = 918 Answer
Similar Questions
(1) Find the average of even numbers from 12 to 644
(2) What is the average of the first 1421 even numbers?
(3) Find the average of the first 3581 odd numbers.
(4) Find the average of the first 2590 odd numbers.
(5) Find the average of the first 799 odd numbers.
(6) Find the average of the first 2062 odd numbers.
(7) What will be the average of the first 4134 odd numbers?
(8) Find the average of the first 1789 odd numbers.
(9) Find the average of even numbers from 4 to 498
(10) Find the average of the first 2573 even numbers.