Question:
Find the average of the first 2093 even numbers.
Correct Answer
2094
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 2093 even numbers are
2, 4, 6, 8, . . . . 2093 th terms
Calculation of the sum of the first 2093 even numbers
We can find the sum of the first 2093 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 2093 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 2093 even number,
n = 2093, a = 2, and d = 2
Thus, sum of the first 2093 even numbers
S2093 = 2093/2 [2 × 2 + (2093 – 1) 2]
= 2093/2 [4 + 2092 × 2]
= 2093/2 [4 + 4184]
= 2093/2 × 4188
= 2093/2 × 4188 2094
= 2093 × 2094 = 4382742
⇒ The sum of the first 2093 even numbers (S2093) = 4382742
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 2093 even numbers
= 20932 + 2093
= 4380649 + 2093 = 4382742
⇒ The sum of the first 2093 even numbers = 4382742
Calculation of the Average of the first 2093 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 2093 even numbers
= Sum of the first 2093 even numbers/2093
= 4382742/2093 = 2094
Thus, the average of the first 2093 even numbers = 2094 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 2093 even numbers = 2093 + 1 = 2094
Thus, the average of the first 2093 even numbers = 2094 Answer
Similar Questions
(1) Find the average of even numbers from 12 to 1820
(2) Find the average of even numbers from 4 to 1622
(3) What is the average of the first 705 even numbers?
(4) What is the average of the first 1713 even numbers?
(5) What is the average of the first 87 even numbers?
(6) Find the average of even numbers from 12 to 1376
(7) What will be the average of the first 4005 odd numbers?
(8) What is the average of the first 1745 even numbers?
(9) What is the average of the first 987 even numbers?
(10) Find the average of the first 801 odd numbers.