Question:
What is the average of the first 1557 even numbers?
Correct Answer
1558
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 1557 even numbers are
2, 4, 6, 8, . . . . 1557 th terms
Calculation of the sum of the first 1557 even numbers
We can find the sum of the first 1557 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 1557 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 1557 even number,
n = 1557, a = 2, and d = 2
Thus, sum of the first 1557 even numbers
S1557 = 1557/2 [2 × 2 + (1557 – 1) 2]
= 1557/2 [4 + 1556 × 2]
= 1557/2 [4 + 3112]
= 1557/2 × 3116
= 1557/2 × 3116 1558
= 1557 × 1558 = 2425806
⇒ The sum of the first 1557 even numbers (S1557) = 2425806
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 1557 even numbers
= 15572 + 1557
= 2424249 + 1557 = 2425806
⇒ The sum of the first 1557 even numbers = 2425806
Calculation of the Average of the first 1557 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 1557 even numbers
= Sum of the first 1557 even numbers/1557
= 2425806/1557 = 1558
Thus, the average of the first 1557 even numbers = 1558 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 1557 even numbers = 1557 + 1 = 1558
Thus, the average of the first 1557 even numbers = 1558 Answer
Similar Questions
(1) What will be the average of the first 4179 odd numbers?
(2) Find the average of the first 2724 odd numbers.
(3) Find the average of odd numbers from 3 to 873
(4) Find the average of odd numbers from 13 to 1031
(5) Find the average of the first 3291 odd numbers.
(6) Find the average of even numbers from 8 to 1016
(7) What will be the average of the first 4508 odd numbers?
(8) What is the average of the first 166 odd numbers?
(9) Find the average of the first 2580 even numbers.
(10) What is the average of the first 98 even numbers?