Question:
What is the average of the first 1006 even numbers?
Correct Answer
1007
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 1006 even numbers are
2, 4, 6, 8, . . . . 1006 th terms
Calculation of the sum of the first 1006 even numbers
We can find the sum of the first 1006 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 1006 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 1006 even number,
n = 1006, a = 2, and d = 2
Thus, sum of the first 1006 even numbers
S1006 = 1006/2 [2 × 2 + (1006 – 1) 2]
= 1006/2 [4 + 1005 × 2]
= 1006/2 [4 + 2010]
= 1006/2 × 2014
= 1006/2 × 2014 1007
= 1006 × 1007 = 1013042
⇒ The sum of the first 1006 even numbers (S1006) = 1013042
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 1006 even numbers
= 10062 + 1006
= 1012036 + 1006 = 1013042
⇒ The sum of the first 1006 even numbers = 1013042
Calculation of the Average of the first 1006 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 1006 even numbers
= Sum of the first 1006 even numbers/1006
= 1013042/1006 = 1007
Thus, the average of the first 1006 even numbers = 1007 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 1006 even numbers = 1006 + 1 = 1007
Thus, the average of the first 1006 even numbers = 1007 Answer
Similar Questions
(1) Find the average of the first 2066 odd numbers.
(2) Find the average of the first 3659 even numbers.
(3) Find the average of odd numbers from 7 to 1167
(4) Find the average of even numbers from 6 to 1524
(5) What is the average of the first 228 even numbers?
(6) Find the average of the first 4087 even numbers.
(7) Find the average of the first 2217 odd numbers.
(8) What is the average of the first 64 odd numbers?
(9) Find the average of the first 1054 odd numbers.
(10) Find the average of odd numbers from 9 to 1079