Question:
Find the average of even numbers from 12 to 842
Correct Answer
427
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 842
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 842 are
12, 14, 16, . . . . 842
After observing the above list of the even numbers from 12 to 842 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 842 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 842
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 842
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 842
= 12 + 842/2
= 854/2 = 427
Thus, the average of the even numbers from 12 to 842 = 427 Answer
Method (2) to find the average of the even numbers from 12 to 842
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 842 are
12, 14, 16, . . . . 842
The even numbers from 12 to 842 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 842
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 842
842 = 12 + (n – 1) × 2
⇒ 842 = 12 + 2 n – 2
⇒ 842 = 12 – 2 + 2 n
⇒ 842 = 10 + 2 n
After transposing 10 to LHS
⇒ 842 – 10 = 2 n
⇒ 832 = 2 n
After rearranging the above expression
⇒ 2 n = 832
After transposing 2 to RHS
⇒ n = 832/2
⇒ n = 416
Thus, the number of terms of even numbers from 12 to 842 = 416
This means 842 is the 416th term.
Finding the sum of the given even numbers from 12 to 842
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 842
= 416/2 (12 + 842)
= 416/2 × 854
= 416 × 854/2
= 355264/2 = 177632
Thus, the sum of all terms of the given even numbers from 12 to 842 = 177632
And, the total number of terms = 416
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 842
= 177632/416 = 427
Thus, the average of the given even numbers from 12 to 842 = 427 Answer
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