Question:
Find the average of even numbers from 12 to 886
Correct Answer
449
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 886
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 886 are
12, 14, 16, . . . . 886
After observing the above list of the even numbers from 12 to 886 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 886 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 886
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 886
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 886
= 12 + 886/2
= 898/2 = 449
Thus, the average of the even numbers from 12 to 886 = 449 Answer
Method (2) to find the average of the even numbers from 12 to 886
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 886 are
12, 14, 16, . . . . 886
The even numbers from 12 to 886 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 886
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 886
886 = 12 + (n – 1) × 2
⇒ 886 = 12 + 2 n – 2
⇒ 886 = 12 – 2 + 2 n
⇒ 886 = 10 + 2 n
After transposing 10 to LHS
⇒ 886 – 10 = 2 n
⇒ 876 = 2 n
After rearranging the above expression
⇒ 2 n = 876
After transposing 2 to RHS
⇒ n = 876/2
⇒ n = 438
Thus, the number of terms of even numbers from 12 to 886 = 438
This means 886 is the 438th term.
Finding the sum of the given even numbers from 12 to 886
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 886
= 438/2 (12 + 886)
= 438/2 × 898
= 438 × 898/2
= 393324/2 = 196662
Thus, the sum of all terms of the given even numbers from 12 to 886 = 196662
And, the total number of terms = 438
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 886
= 196662/438 = 449
Thus, the average of the given even numbers from 12 to 886 = 449 Answer
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