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Question:     Find the average of odd numbers from 15 to 1473


Correct Answer  744

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 15 to 1473

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

The odd numbers from 15 to 1473 are

15, 17, 19, . . . . 1473

After observing the above list of the odd numbers from 15 to 1473 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1473 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 15 to 1473

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 1473

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 odd numbers from 15 to 1473

= 15 + 1473/2

= 1488/2 = 744

Thus, the average of the odd numbers from 15 to 1473 = 744 Answer

Method (2) to find the average of the odd numbers from 15 to 1473

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

The odd numbers from 15 to 1473 are

15, 17, 19, . . . . 1473

The odd numbers from 15 to 1473 form an Arithmetic Series in which

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 1473

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 odd numbers from 15 to 1473

1473 = 15 + (n – 1) × 2

⇒ 1473 = 15 + 2 n – 2

⇒ 1473 = 15 – 2 + 2 n

⇒ 1473 = 13 + 2 n

After transposing 13 to LHS

⇒ 1473 – 13 = 2 n

⇒ 1460 = 2 n

After rearranging the above expression

⇒ 2 n = 1460

After transposing 2 to RHS

⇒ n = 1460/2

⇒ n = 730

Thus, the number of terms of odd numbers from 15 to 1473 = 730

This means 1473 is the 730th term.

Finding the sum of the given odd numbers from 15 to 1473

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 odd numbers from 15 to 1473

= 730/2 (15 + 1473)

= 730/2 × 1488

= 730 × 1488/2

= 1086240/2 = 543120

Thus, the sum of all terms of the given odd numbers from 15 to 1473 = 543120

And, the total number of terms = 730

Since, the average of the given numbers

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

Thus, the average of the given odd numbers from 15 to 1473

= 543120/730 = 744

Thus, the average of the given odd numbers from 15 to 1473 = 744 Answer


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