Question:
Find the average of even numbers from 6 to 1014
Correct Answer
510
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1014
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1014 are
6, 8, 10, . . . . 1014
After observing the above list of the even numbers from 6 to 1014 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1014 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1014
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1014
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 6 to 1014
= 6 + 1014/2
= 1020/2 = 510
Thus, the average of the even numbers from 6 to 1014 = 510 Answer
Method (2) to find the average of the even numbers from 6 to 1014
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1014 are
6, 8, 10, . . . . 1014
The even numbers from 6 to 1014 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1014
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 6 to 1014
1014 = 6 + (n – 1) × 2
⇒ 1014 = 6 + 2 n – 2
⇒ 1014 = 6 – 2 + 2 n
⇒ 1014 = 4 + 2 n
After transposing 4 to LHS
⇒ 1014 – 4 = 2 n
⇒ 1010 = 2 n
After rearranging the above expression
⇒ 2 n = 1010
After transposing 2 to RHS
⇒ n = 1010/2
⇒ n = 505
Thus, the number of terms of even numbers from 6 to 1014 = 505
This means 1014 is the 505th term.
Finding the sum of the given even numbers from 6 to 1014
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 6 to 1014
= 505/2 (6 + 1014)
= 505/2 × 1020
= 505 × 1020/2
= 515100/2 = 257550
Thus, the sum of all terms of the given even numbers from 6 to 1014 = 257550
And, the total number of terms = 505
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 6 to 1014
= 257550/505 = 510
Thus, the average of the given even numbers from 6 to 1014 = 510 Answer
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