Question : What is the average of the first 1238 even numbers?
Correct Answer 1239
Solution & 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 1238 even numbers are
2, 4, 6, 8, . . . . 1238 th terms
Calculation of the sum of the first 1238 even numbers
We can find the sum of the first 1238 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 1238 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 1238 even number,
n = 1238, a = 2, and d = 2
Thus, sum of the first 1238 even numbers
S1238 = 1238/2 [2 × 2 + (1238 – 1) 2]
= 1238/2 [4 + 1237 × 2]
= 1238/2 [4 + 2474]
= 1238/2 × 2478
= 1238/2 × 2478 1239
= 1238 × 1239 = 1533882
⇒ The sum of the first 1238 even numbers (S1238) = 1533882
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 1238 even numbers
= 12382 + 1238
= 1532644 + 1238 = 1533882
⇒ The sum of the first 1238 even numbers = 1533882
Calculation of the Average of the first 1238 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 1238 even numbers
= Sum of the first 1238 even numbers/1238
= 1533882/1238 = 1239
Thus, the average of the first 1238 even numbers = 1239 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 1238 even numbers = 1238 + 1 = 1239
Thus, the average of the first 1238 even numbers = 1239 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1902
(2) Find the average of the first 3955 even numbers.
(3) Find the average of odd numbers from 5 to 1041
(4) Find the average of even numbers from 10 to 1520
(5) Find the average of the first 3113 even numbers.
(6) Find the average of odd numbers from 15 to 881
(7) Find the average of odd numbers from 5 to 37
(8) What is the average of the first 642 even numbers?
(9) What is the average of the first 37 even numbers?
(10) Find the average of the first 4185 even numbers.