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Question :    Find the average of even numbers from 12 to 1406


Correct Answer  709

Solution & Explanation

Solution

Method (1) to find the average of the even numbers from 12 to 1406

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

The even numbers from 12 to 1406 are

12, 14, 16, . . . . 1406

After observing the above list of the even numbers from 12 to 1406 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1406 form an Arithmetic Series.

In the Arithmetic Series of the even numbers from 12 to 1406

The First Term (a) = 12

The Common Difference (d) = 2

And the last term (ℓ) = 1406

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 12 to 1406

= 12 + 1406/2

= 1418/2 = 709

Thus, the average of the even numbers from 12 to 1406 = 709 Answer

Method (2) to find the average of the even numbers from 12 to 1406

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

The even numbers from 12 to 1406 are

12, 14, 16, . . . . 1406

The even numbers from 12 to 1406 form an Arithmetic Series in which

The First Term (a) = 12

The Common Difference (d) = 2

And the last term (ℓ) = 1406

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 12 to 1406

1406 = 12 + (n – 1) × 2

⇒ 1406 = 12 + 2 n – 2

⇒ 1406 = 12 – 2 + 2 n

⇒ 1406 = 10 + 2 n

After transposing 10 to LHS

⇒ 1406 – 10 = 2 n

⇒ 1396 = 2 n

After rearranging the above expression

⇒ 2 n = 1396

After transposing 2 to RHS

⇒ n = 1396/2

⇒ n = 698

Thus, the number of terms of even numbers from 12 to 1406 = 698

This means 1406 is the 698th term.

Finding the sum of the given even numbers from 12 to 1406

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 12 to 1406

= 698/2 (12 + 1406)

= 698/2 × 1418

= 698 × 1418/2

= 989764/2 = 494882

Thus, the sum of all terms of the given even numbers from 12 to 1406 = 494882

And, the total number of terms = 698

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 12 to 1406

= 494882/698 = 709

Thus, the average of the given even numbers from 12 to 1406 = 709 Answer


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