Question:
Find the average of the first 4096 even numbers.
Correct Answer
4097
Solution And Explanation
Explanation
Method to find the average
Step : (1) Find the sum of given numbers
Step: (2) Divide the sum of given number by the number of numbers. This will give the average of the given numbers
The first 4096 even numbers are
2, 4, 6, 8, . . . . 4096 th terms
Calculation of the sum of the first 4096 even numbers
We can find the sum of the first 4096 even numbers by simply adding them, but this is a bit difficult. And if the list is long, it is very difficult to find their sum. So, in such a situation, we will use a formula to find the sum of given numbers that form a particular pattern.
Here, the list of the first 4096 even numbers forms an Arithmetic series
In an Arithmetic Series, the common difference is the same. This means the difference between two consecutive terms are same in an Arithmetic Series.
The sum of n terms of an Arithmetic Series
Sn = n/2 [2a + (n – 1) d]
Where, n = number of terms, a = first term, and d = common difference
In the series of the first 4096 even number,
n = 4096, a = 2, and d = 2
Thus, sum of the first 4096 even numbers
S4096 = 4096/2 [2 × 2 + (4096 – 1) 2]
= 4096/2 [4 + 4095 × 2]
= 4096/2 [4 + 8190]
= 4096/2 × 8194
= 4096/2 × 8194 4097
= 4096 × 4097 = 16781312
⇒ The sum of the first 4096 even numbers (S4096) = 16781312
Shortcut Method to find the sum of the first n even numbers
Thus, the sum of the first n even numbers = n2 + n
Thus, the sum of the first 4096 even numbers
= 40962 + 4096
= 16777216 + 4096 = 16781312
⇒ The sum of the first 4096 even numbers = 16781312
Calculation of the Average of the first 4096 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 4096 even numbers
= Sum of the first 4096 even numbers/4096
= 16781312/4096 = 4097
Thus, the average of the first 4096 even numbers = 4097 Answer
Shortcut Trick to find the Average of the first n even numbers
(1) The average of the first 2 even numbers
= 2 + 4/2
= 6/2 = 3
Thus, the average of the first 2 even numbers = 3
(2) The average of the first 3 even numbers
= 2 + 4 + 6/3
= 12/3 = 4
Thus, the average of the first 3 even numbers = 4
(3) The average of the first 4 even numbers
= 2 + 4 + 6 + 8/4
= 20/4 = 5
Thus, the average of the first 4 even numbers = 5
(4) The average of the first 5 even numbers
= 2 + 4 + 6 + 8 + 10/5
= 30/5 = 6
Thus, the average of the first 5 even numbers = 6
Thus, the Average of the First n even numbers = n + 1
Thus, the average of the first 4096 even numbers = 4096 + 1 = 4097
Thus, the average of the first 4096 even numbers = 4097 Answer
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