Average
MCQs Math


Question:     Find the average of the first 2234 even numbers.


Correct Answer  2235

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 2234 even numbers are

2, 4, 6, 8, . . . . 2234 th terms

Calculation of the sum of the first 2234 even numbers

We can find the sum of the first 2234 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 2234 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 2234 even number,

n = 2234, a = 2, and d = 2

Thus, sum of the first 2234 even numbers

S2234 = 2234/2 [2 × 2 + (2234 – 1) 2]

= 2234/2 [4 + 2233 × 2]

= 2234/2 [4 + 4466]

= 2234/2 × 4470

= 2234/2 × 4470 2235

= 2234 × 2235 = 4992990

⇒ The sum of the first 2234 even numbers (S2234) = 4992990

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 2234 even numbers

= 22342 + 2234

= 4990756 + 2234 = 4992990

⇒ The sum of the first 2234 even numbers = 4992990

Calculation of the Average of the first 2234 even numbers

Formula to find the Average

Average = Sum of the given numbers/Number of the numbers

Thus, The average of the first 2234 even numbers

= Sum of the first 2234 even numbers/2234

= 4992990/2234 = 2235

Thus, the average of the first 2234 even numbers = 2235 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 2234 even numbers = 2234 + 1 = 2235

Thus, the average of the first 2234 even numbers = 2235 Answer


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