Question : Find the average of the first 4962 even numbers.
Correct Answer 4963
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 4962 even numbers are
2, 4, 6, 8, . . . . 4962 th terms
Calculation of the sum of the first 4962 even numbers
We can find the sum of the first 4962 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 4962 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 4962 even number,
n = 4962, a = 2, and d = 2
Thus, sum of the first 4962 even numbers
S4962 = 4962/2 [2 × 2 + (4962 – 1) 2]
= 4962/2 [4 + 4961 × 2]
= 4962/2 [4 + 9922]
= 4962/2 × 9926
= 4962/2 × 9926 4963
= 4962 × 4963 = 24626406
⇒ The sum of the first 4962 even numbers (S4962) = 24626406
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 4962 even numbers
= 49622 + 4962
= 24621444 + 4962 = 24626406
⇒ The sum of the first 4962 even numbers = 24626406
Calculation of the Average of the first 4962 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 4962 even numbers
= Sum of the first 4962 even numbers/4962
= 24626406/4962 = 4963
Thus, the average of the first 4962 even numbers = 4963 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 4962 even numbers = 4962 + 1 = 4963
Thus, the average of the first 4962 even numbers = 4963 Answer
Similar Questions
(1) Find the average of odd numbers from 11 to 697
(2) Find the average of odd numbers from 5 to 1457
(3) Find the average of the first 3748 odd numbers.
(4) Find the average of odd numbers from 11 to 1187
(5) What is the average of the first 1040 even numbers?
(6) Find the average of even numbers from 12 to 1152
(7) Find the average of even numbers from 6 to 1690
(8) Find the average of even numbers from 4 to 1354
(9) Find the average of odd numbers from 15 to 1473
(10) Find the average of odd numbers from 11 to 479