Question:
Find the average of the first 3070 even numbers.
Correct Answer
3071
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 3070 even numbers are
2, 4, 6, 8, . . . . 3070 th terms
Calculation of the sum of the first 3070 even numbers
We can find the sum of the first 3070 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 3070 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 3070 even number,
n = 3070, a = 2, and d = 2
Thus, sum of the first 3070 even numbers
S3070 = 3070/2 [2 × 2 + (3070 – 1) 2]
= 3070/2 [4 + 3069 × 2]
= 3070/2 [4 + 6138]
= 3070/2 × 6142
= 3070/2 × 6142 3071
= 3070 × 3071 = 9427970
⇒ The sum of the first 3070 even numbers (S3070) = 9427970
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 3070 even numbers
= 30702 + 3070
= 9424900 + 3070 = 9427970
⇒ The sum of the first 3070 even numbers = 9427970
Calculation of the Average of the first 3070 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 3070 even numbers
= Sum of the first 3070 even numbers/3070
= 9427970/3070 = 3071
Thus, the average of the first 3070 even numbers = 3071 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 3070 even numbers = 3070 + 1 = 3071
Thus, the average of the first 3070 even numbers = 3071 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 687
(2) Find the average of the first 2472 even numbers.
(3) What will be the average of the first 4597 odd numbers?
(4) What is the average of the first 1828 even numbers?
(5) What is the average of the first 1137 even numbers?
(6) Find the average of the first 3003 odd numbers.
(7) Find the average of odd numbers from 5 to 789
(8) Find the average of the first 3373 odd numbers.
(9) Find the average of the first 2901 odd numbers.
(10) Find the average of even numbers from 12 to 1004