Question:
What is the average of the first 1289 even numbers?
Correct Answer
1290
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 1289 even numbers are
2, 4, 6, 8, . . . . 1289 th terms
Calculation of the sum of the first 1289 even numbers
We can find the sum of the first 1289 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 1289 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 1289 even number,
n = 1289, a = 2, and d = 2
Thus, sum of the first 1289 even numbers
S1289 = 1289/2 [2 × 2 + (1289 – 1) 2]
= 1289/2 [4 + 1288 × 2]
= 1289/2 [4 + 2576]
= 1289/2 × 2580
= 1289/2 × 2580 1290
= 1289 × 1290 = 1662810
⇒ The sum of the first 1289 even numbers (S1289) = 1662810
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 1289 even numbers
= 12892 + 1289
= 1661521 + 1289 = 1662810
⇒ The sum of the first 1289 even numbers = 1662810
Calculation of the Average of the first 1289 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 1289 even numbers
= Sum of the first 1289 even numbers/1289
= 1662810/1289 = 1290
Thus, the average of the first 1289 even numbers = 1290 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 1289 even numbers = 1289 + 1 = 1290
Thus, the average of the first 1289 even numbers = 1290 Answer
Similar Questions
(1) What is the average of the first 12 odd numbers?
(2) Find the average of even numbers from 10 to 1840
(3) Find the average of the first 3887 odd numbers.
(4) What will be the average of the first 4783 odd numbers?
(5) Find the average of the first 2670 even numbers.
(6) Find the average of the first 2809 even numbers.
(7) Find the average of the first 945 odd numbers.
(8) Find the average of odd numbers from 13 to 1221
(9) Find the average of odd numbers from 13 to 755
(10) Find the average of the first 4990 even numbers.