Question:
Find the average of the first 2198 even numbers.
Correct Answer
2199
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 2198 even numbers are
2, 4, 6, 8, . . . . 2198 th terms
Calculation of the sum of the first 2198 even numbers
We can find the sum of the first 2198 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 2198 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 2198 even number,
n = 2198, a = 2, and d = 2
Thus, sum of the first 2198 even numbers
S2198 = 2198/2 [2 × 2 + (2198 – 1) 2]
= 2198/2 [4 + 2197 × 2]
= 2198/2 [4 + 4394]
= 2198/2 × 4398
= 2198/2 × 4398 2199
= 2198 × 2199 = 4833402
⇒ The sum of the first 2198 even numbers (S2198) = 4833402
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 2198 even numbers
= 21982 + 2198
= 4831204 + 2198 = 4833402
⇒ The sum of the first 2198 even numbers = 4833402
Calculation of the Average of the first 2198 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 2198 even numbers
= Sum of the first 2198 even numbers/2198
= 4833402/2198 = 2199
Thus, the average of the first 2198 even numbers = 2199 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 2198 even numbers = 2198 + 1 = 2199
Thus, the average of the first 2198 even numbers = 2199 Answer
Similar Questions
(1) Find the average of the first 2319 odd numbers.
(2) Find the average of odd numbers from 13 to 1249
(3) Find the average of the first 3050 odd numbers.
(4) Find the average of even numbers from 6 to 1328
(5) Find the average of odd numbers from 11 to 1057
(6) Find the average of the first 1277 odd numbers.
(7) Find the average of the first 3730 even numbers.
(8) Find the average of odd numbers from 7 to 691
(9) Find the average of odd numbers from 5 to 589
(10) Find the average of even numbers from 10 to 1208