Question : Find the average of the first 2207 even numbers.
Correct Answer 2208
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 2207 even numbers are
2, 4, 6, 8, . . . . 2207 th terms
Calculation of the sum of the first 2207 even numbers
We can find the sum of the first 2207 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 2207 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 2207 even number,
n = 2207, a = 2, and d = 2
Thus, sum of the first 2207 even numbers
S2207 = 2207/2 [2 × 2 + (2207 – 1) 2]
= 2207/2 [4 + 2206 × 2]
= 2207/2 [4 + 4412]
= 2207/2 × 4416
= 2207/2 × 4416 2208
= 2207 × 2208 = 4873056
⇒ The sum of the first 2207 even numbers (S2207) = 4873056
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 2207 even numbers
= 22072 + 2207
= 4870849 + 2207 = 4873056
⇒ The sum of the first 2207 even numbers = 4873056
Calculation of the Average of the first 2207 even numbers
Formula to find the Average
Average = Sum of the given numbers/Number of the numbers
Thus, The average of the first 2207 even numbers
= Sum of the first 2207 even numbers/2207
= 4873056/2207 = 2208
Thus, the average of the first 2207 even numbers = 2208 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 2207 even numbers = 2207 + 1 = 2208
Thus, the average of the first 2207 even numbers = 2208 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 1231
(2) Find the average of the first 3602 odd numbers.
(3) What is the average of the first 864 even numbers?
(4) Find the average of the first 1153 odd numbers.
(5) Find the average of the first 1175 odd numbers.
(6) Find the average of even numbers from 12 to 1830
(7) Find the average of the first 4124 even numbers.
(8) Find the average of odd numbers from 7 to 1117
(9) Find the average of the first 3360 odd numbers.
(10) Find the average of the first 3225 even numbers.