Question:
Find the average of the first 3193 even numbers.
Correct Answer
3194
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 3193 even numbers are
2, 4, 6, 8, . . . . 3193 th terms
Calculation of the sum of the first 3193 even numbers
We can find the sum of the first 3193 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 3193 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 3193 even number,
n = 3193, a = 2, and d = 2
Thus, sum of the first 3193 even numbers
S3193 = 3193/2 [2 × 2 + (3193 – 1) 2]
= 3193/2 [4 + 3192 × 2]
= 3193/2 [4 + 6384]
= 3193/2 × 6388
= 3193/2 × 6388 3194
= 3193 × 3194 = 10198442
⇒ The sum of the first 3193 even numbers (S3193) = 10198442
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 3193 even numbers
= 31932 + 3193
= 10195249 + 3193 = 10198442
⇒ The sum of the first 3193 even numbers = 10198442
Calculation of the Average of the first 3193 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 3193 even numbers
= Sum of the first 3193 even numbers/3193
= 10198442/3193 = 3194
Thus, the average of the first 3193 even numbers = 3194 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 3193 even numbers = 3193 + 1 = 3194
Thus, the average of the first 3193 even numbers = 3194 Answer
Similar Questions
(1) Find the average of the first 1439 odd numbers.
(2) Find the average of even numbers from 12 to 82
(3) Find the average of the first 4200 even numbers.
(4) Find the average of even numbers from 10 to 948
(5) Find the average of the first 1586 odd numbers.
(6) Find the average of even numbers from 10 to 232
(7) What will be the average of the first 4036 odd numbers?
(8) Find the average of the first 3859 even numbers.
(9) Find the average of the first 3202 even numbers.
(10) Find the average of even numbers from 10 to 2000