Question:
Find the average of even numbers from 12 to 884
Correct Answer
448
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 884
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 884 are
12, 14, 16, . . . . 884
After observing the above list of the even numbers from 12 to 884 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 884 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 884
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 884
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 884
= 12 + 884/2
= 896/2 = 448
Thus, the average of the even numbers from 12 to 884 = 448 Answer
Method (2) to find the average of the even numbers from 12 to 884
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 884 are
12, 14, 16, . . . . 884
The even numbers from 12 to 884 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 884
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 884
884 = 12 + (n – 1) × 2
⇒ 884 = 12 + 2 n – 2
⇒ 884 = 12 – 2 + 2 n
⇒ 884 = 10 + 2 n
After transposing 10 to LHS
⇒ 884 – 10 = 2 n
⇒ 874 = 2 n
After rearranging the above expression
⇒ 2 n = 874
After transposing 2 to RHS
⇒ n = 874/2
⇒ n = 437
Thus, the number of terms of even numbers from 12 to 884 = 437
This means 884 is the 437th term.
Finding the sum of the given even numbers from 12 to 884
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 884
= 437/2 (12 + 884)
= 437/2 × 896
= 437 × 896/2
= 391552/2 = 195776
Thus, the sum of all terms of the given even numbers from 12 to 884 = 195776
And, the total number of terms = 437
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 884
= 195776/437 = 448
Thus, the average of the given even numbers from 12 to 884 = 448 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 1377
(2) What is the average of the first 551 even numbers?
(3) Find the average of the first 1638 odd numbers.
(4) Find the average of the first 3576 even numbers.
(5) What will be the average of the first 4063 odd numbers?
(6) Find the average of the first 2021 odd numbers.
(7) Find the average of odd numbers from 11 to 1323
(8) Find the average of even numbers from 10 to 1052
(9) Find the average of the first 1448 odd numbers.
(10) What is the average of the first 912 even numbers?