Question : Find the average of even numbers from 4 to 1028
Correct Answer 516
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1028
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1028 are
4, 6, 8, . . . . 1028
After observing the above list of the even numbers from 4 to 1028 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1028 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1028
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1028
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 4 to 1028
= 4 + 1028/2
= 1032/2 = 516
Thus, the average of the even numbers from 4 to 1028 = 516 Answer
Method (2) to find the average of the even numbers from 4 to 1028
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1028 are
4, 6, 8, . . . . 1028
The even numbers from 4 to 1028 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1028
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 4 to 1028
1028 = 4 + (n – 1) × 2
⇒ 1028 = 4 + 2 n – 2
⇒ 1028 = 4 – 2 + 2 n
⇒ 1028 = 2 + 2 n
After transposing 2 to LHS
⇒ 1028 – 2 = 2 n
⇒ 1026 = 2 n
After rearranging the above expression
⇒ 2 n = 1026
After transposing 2 to RHS
⇒ n = 1026/2
⇒ n = 513
Thus, the number of terms of even numbers from 4 to 1028 = 513
This means 1028 is the 513th term.
Finding the sum of the given even numbers from 4 to 1028
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 4 to 1028
= 513/2 (4 + 1028)
= 513/2 × 1032
= 513 × 1032/2
= 529416/2 = 264708
Thus, the sum of all terms of the given even numbers from 4 to 1028 = 264708
And, the total number of terms = 513
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 4 to 1028
= 264708/513 = 516
Thus, the average of the given even numbers from 4 to 1028 = 516 Answer
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