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Question:     Find the average of even numbers from 6 to 1652


Correct Answer  829

Solution And Explanation

Solution

Method (1) to find the average of the even numbers from 6 to 1652

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

The even numbers from 6 to 1652 are

6, 8, 10, . . . . 1652

After observing the above list of the even numbers from 6 to 1652 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1652 form an Arithmetic Series.

In the Arithmetic Series of the even numbers from 6 to 1652

The First Term (a) = 6

The Common Difference (d) = 2

And the last term (ℓ) = 1652

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 6 to 1652

= 6 + 1652/2

= 1658/2 = 829

Thus, the average of the even numbers from 6 to 1652 = 829 Answer

Method (2) to find the average of the even numbers from 6 to 1652

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

The even numbers from 6 to 1652 are

6, 8, 10, . . . . 1652

The even numbers from 6 to 1652 form an Arithmetic Series in which

The First Term (a) = 6

The Common Difference (d) = 2

And the last term (ℓ) = 1652

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 6 to 1652

1652 = 6 + (n – 1) × 2

⇒ 1652 = 6 + 2 n – 2

⇒ 1652 = 6 – 2 + 2 n

⇒ 1652 = 4 + 2 n

After transposing 4 to LHS

⇒ 1652 – 4 = 2 n

⇒ 1648 = 2 n

After rearranging the above expression

⇒ 2 n = 1648

After transposing 2 to RHS

⇒ n = 1648/2

⇒ n = 824

Thus, the number of terms of even numbers from 6 to 1652 = 824

This means 1652 is the 824th term.

Finding the sum of the given even numbers from 6 to 1652

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 6 to 1652

= 824/2 (6 + 1652)

= 824/2 × 1658

= 824 × 1658/2

= 1366192/2 = 683096

Thus, the sum of all terms of the given even numbers from 6 to 1652 = 683096

And, the total number of terms = 824

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 6 to 1652

= 683096/824 = 829

Thus, the average of the given even numbers from 6 to 1652 = 829 Answer


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