Question:
Find the average of the first 2972 even numbers.
Correct Answer
2973
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 2972 even numbers are
2, 4, 6, 8, . . . . 2972 th terms
Calculation of the sum of the first 2972 even numbers
We can find the sum of the first 2972 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 2972 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 2972 even number,
n = 2972, a = 2, and d = 2
Thus, sum of the first 2972 even numbers
S2972 = 2972/2 [2 × 2 + (2972 – 1) 2]
= 2972/2 [4 + 2971 × 2]
= 2972/2 [4 + 5942]
= 2972/2 × 5946
= 2972/2 × 5946 2973
= 2972 × 2973 = 8835756
⇒ The sum of the first 2972 even numbers (S2972) = 8835756
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 2972 even numbers
= 29722 + 2972
= 8832784 + 2972 = 8835756
⇒ The sum of the first 2972 even numbers = 8835756
Calculation of the Average of the first 2972 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 2972 even numbers
= Sum of the first 2972 even numbers/2972
= 8835756/2972 = 2973
Thus, the average of the first 2972 even numbers = 2973 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 2972 even numbers = 2972 + 1 = 2973
Thus, the average of the first 2972 even numbers = 2973 Answer
Similar Questions
(1) What is the average of the first 1290 even numbers?
(2) Find the average of the first 2109 even numbers.
(3) Find the average of the first 3274 even numbers.
(4) Find the average of odd numbers from 11 to 545
(5) Find the average of the first 2106 even numbers.
(6) Find the average of the first 2047 odd numbers.
(7) Find the average of odd numbers from 5 to 1043
(8) Find the average of the first 1869 odd numbers.
(9) Find the average of the first 1675 odd numbers.
(10) Find the average of odd numbers from 9 to 425