Question:
Find the average of the first 2047 even numbers.
Correct Answer
2048
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 2047 even numbers are
2, 4, 6, 8, . . . . 2047 th terms
Calculation of the sum of the first 2047 even numbers
We can find the sum of the first 2047 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 2047 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 2047 even number,
n = 2047, a = 2, and d = 2
Thus, sum of the first 2047 even numbers
S2047 = 2047/2 [2 × 2 + (2047 – 1) 2]
= 2047/2 [4 + 2046 × 2]
= 2047/2 [4 + 4092]
= 2047/2 × 4096
= 2047/2 × 4096 2048
= 2047 × 2048 = 4192256
⇒ The sum of the first 2047 even numbers (S2047) = 4192256
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 2047 even numbers
= 20472 + 2047
= 4190209 + 2047 = 4192256
⇒ The sum of the first 2047 even numbers = 4192256
Calculation of the Average of the first 2047 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 2047 even numbers
= Sum of the first 2047 even numbers/2047
= 4192256/2047 = 2048
Thus, the average of the first 2047 even numbers = 2048 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 2047 even numbers = 2047 + 1 = 2048
Thus, the average of the first 2047 even numbers = 2048 Answer
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