🏡 Home
    1. Time and Distance
    2. Time and Work
    3. Profit And Loss
    4. Average
    5. Percentage
    6. Simple Interest
    7. Questions based on ages
    1. Math
    2. Chemistry
    3. Chemistry Hindi
    4. Biology
    5. Exemplar Solution
    1. 11th physics
    2. 11th physics-hindi
    1. Science 10th (English)
    2. Science 10th (Hindi)
    3. Mathematics
    4. Math (Hindi)
    5. Social Science
    1. Science (English)
    2. 9th-Science (Hindi)
    1. 8th-Science (English)
    2. 8th-Science (Hindi)
    3. 8th-math (English)
    4. 8th-math (Hindi)
    1. 7th Math
    2. 7th Math(Hindi)
    1. Sixth Science
    2. 6th Science(hindi)
    1. Five Science
    1. Science (English)
    2. Science (Hindi)
    1. Std 10 science
    2. Std 4 science
    3. Std two EVS
    4. Std two Math
    5. MCQs Math
    6. एमoसीoक्यूo गणित
    7. Civil Service
    1. General Math (Hindi version)
    1. About Us
    2. Contact Us
10upon10.com

Average
Math MCQs


Question :    Find the average of the first 2014 even numbers.


Correct Answer  2015

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 2014 even numbers are

2, 4, 6, 8, . . . . 2014 th terms

Calculation of the sum of the first 2014 even numbers

We can find the sum of the first 2014 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 2014 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 2014 even number,

n = 2014, a = 2, and d = 2

Thus, sum of the first 2014 even numbers

S2014 = 2014/2 [2 × 2 + (2014 – 1) 2]

= 2014/2 [4 + 2013 × 2]

= 2014/2 [4 + 4026]

= 2014/2 × 4030

= 2014/2 × 4030 2015

= 2014 × 2015 = 4058210

⇒ The sum of the first 2014 even numbers (S2014) = 4058210

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 2014 even numbers

= 20142 + 2014

= 4056196 + 2014 = 4058210

⇒ The sum of the first 2014 even numbers = 4058210

Calculation of the Average of the first 2014 even numbers

Formula to find the Average

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

Thus, The average of the first 2014 even numbers

= Sum of the first 2014 even numbers/2014

= 4058210/2014 = 2015

Thus, the average of the first 2014 even numbers = 2015 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 2014 even numbers = 2014 + 1 = 2015

Thus, the average of the first 2014 even numbers = 2015 Answer


Similar Questions

(1) Find the average of the first 3900 odd numbers.

(2) Find the average of the first 2999 odd numbers.

(3) Find the average of the first 2506 odd numbers.

(4) Find the average of odd numbers from 13 to 975

(5) What will be the average of the first 4800 odd numbers?

(6) Find the average of the first 3361 even numbers.

(7) What will be the average of the first 4095 odd numbers?

(8) Find the average of even numbers from 6 to 554

(9) Find the average of odd numbers from 11 to 539

(10) Find the average of odd numbers from 5 to 1013