Average
MCQs Math


Question:     Find the average of even numbers from 4 to 1924


Correct Answer  964

Solution And Explanation

Solution

Method (1) to find the average of the even numbers from 4 to 1924

Shortcut Trick to find the average of the given continuous even numbers

The even numbers from 4 to 1924 are

4, 6, 8, . . . . 1924

After observing the above list of the even numbers from 4 to 1924 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1924 form an Arithmetic Series.

In the Arithmetic Series of the even numbers from 4 to 1924

The First Term (a) = 4

The Common Difference (d) = 2

And the last term (ℓ) = 1924

The average of the numbers forming an Arithmetic Series

= The first term (a) + The last term (ℓ)/2

⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2

Thus, the average of the even numbers from 4 to 1924

= 4 + 1924/2

= 1928/2 = 964

Thus, the average of the even numbers from 4 to 1924 = 964 Answer

Method (2) to find the average of the even numbers from 4 to 1924

Finding the average of given continuous even numbers after finding their sum

The even numbers from 4 to 1924 are

4, 6, 8, . . . . 1924

The even numbers from 4 to 1924 form an Arithmetic Series in which

The First Term (a) = 4

The Common Difference (d) = 2

And the last term (ℓ) = 1924

The Average of the given numbers

= Sum of the given numbers/Total number of given numbers

Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers

Finding the number of terms

For an Arithmetic Series, the nth term

an = a + (n – 1) d

Where

a = First term

d = Common difference

n = number of terms

an = nth term

Thus, for the given series of the even numbers from 4 to 1924

1924 = 4 + (n – 1) × 2

⇒ 1924 = 4 + 2 n – 2

⇒ 1924 = 4 – 2 + 2 n

⇒ 1924 = 2 + 2 n

After transposing 2 to LHS

⇒ 1924 – 2 = 2 n

⇒ 1922 = 2 n

After rearranging the above expression

⇒ 2 n = 1922

After transposing 2 to RHS

⇒ n = 1922/2

⇒ n = 961

Thus, the number of terms of even numbers from 4 to 1924 = 961

This means 1924 is the 961th term.

Finding the sum of the given even numbers from 4 to 1924

The sum of all terms (S) in an Arithmetic Series

= n/2 (a + ℓ)

Where, n = number of terms

a = First term

And, ℓ = Last term

Thus, the sum of all terms (S) of the given even numbers from 4 to 1924

= 961/2 (4 + 1924)

= 961/2 × 1928

= 961 × 1928/2

= 1852808/2 = 926404

Thus, the sum of all terms of the given even numbers from 4 to 1924 = 926404

And, the total number of terms = 961

Since, the average of the given numbers

= Sum of the given numbers/Total number of given numbers

Thus, the average of the given even numbers from 4 to 1924

= 926404/961 = 964

Thus, the average of the given even numbers from 4 to 1924 = 964 Answer


Similar Questions

(1) What is the average of the first 372 even numbers?

(2) If the average of three consecutive odd numbers is 23, then which is the greatest among these odd numbers?

(3) Find the average of even numbers from 4 to 1400

(4) Find the average of the first 779 odd numbers.

(5) Find the average of the first 4893 even numbers.

(6) Find the average of the first 3812 odd numbers.

(7) Find the average of odd numbers from 11 to 121

(8) What will be the average of the first 4944 odd numbers?

(9) Find the average of odd numbers from 3 to 1239

(10) What is the average of the first 1462 even numbers?


NCERT Solution and CBSE Notes for class twelve, eleventh, tenth, ninth, seventh, sixth, fifth, fourth and General Math for competitive Exams. ©