Question:
Find the average of even numbers from 12 to 1994
Correct Answer
1003
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1994
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1994 are
12, 14, 16, . . . . 1994
After observing the above list of the even numbers from 12 to 1994 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1994 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1994
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1994
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 1994
= 12 + 1994/2
= 2006/2 = 1003
Thus, the average of the even numbers from 12 to 1994 = 1003 Answer
Method (2) to find the average of the even numbers from 12 to 1994
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1994 are
12, 14, 16, . . . . 1994
The even numbers from 12 to 1994 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1994
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 1994
1994 = 12 + (n – 1) × 2
⇒ 1994 = 12 + 2 n – 2
⇒ 1994 = 12 – 2 + 2 n
⇒ 1994 = 10 + 2 n
After transposing 10 to LHS
⇒ 1994 – 10 = 2 n
⇒ 1984 = 2 n
After rearranging the above expression
⇒ 2 n = 1984
After transposing 2 to RHS
⇒ n = 1984/2
⇒ n = 992
Thus, the number of terms of even numbers from 12 to 1994 = 992
This means 1994 is the 992th term.
Finding the sum of the given even numbers from 12 to 1994
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 1994
= 992/2 (12 + 1994)
= 992/2 × 2006
= 992 × 2006/2
= 1989952/2 = 994976
Thus, the sum of all terms of the given even numbers from 12 to 1994 = 994976
And, the total number of terms = 992
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 1994
= 994976/992 = 1003
Thus, the average of the given even numbers from 12 to 1994 = 1003 Answer
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