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Question :    Find the average of odd numbers from 15 to 1363


Correct Answer  689

Solution & Explanation

Solution

Method (1) to find the average of the odd numbers from 15 to 1363

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

The odd numbers from 15 to 1363 are

15, 17, 19, . . . . 1363

After observing the above list of the odd numbers from 15 to 1363 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1363 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 15 to 1363

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 1363

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 1363

= 15 + 1363/2

= 1378/2 = 689

Thus, the average of the odd numbers from 15 to 1363 = 689 Answer

Method (2) to find the average of the odd numbers from 15 to 1363

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

The odd numbers from 15 to 1363 are

15, 17, 19, . . . . 1363

The odd numbers from 15 to 1363 form an Arithmetic Series in which

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 1363

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 1363

1363 = 15 + (n – 1) × 2

⇒ 1363 = 15 + 2 n – 2

⇒ 1363 = 15 – 2 + 2 n

⇒ 1363 = 13 + 2 n

After transposing 13 to LHS

⇒ 1363 – 13 = 2 n

⇒ 1350 = 2 n

After rearranging the above expression

⇒ 2 n = 1350

After transposing 2 to RHS

⇒ n = 1350/2

⇒ n = 675

Thus, the number of terms of odd numbers from 15 to 1363 = 675

This means 1363 is the 675th term.

Finding the sum of the given odd numbers from 15 to 1363

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 1363

= 675/2 (15 + 1363)

= 675/2 × 1378

= 675 × 1378/2

= 930150/2 = 465075

Thus, the sum of all terms of the given odd numbers from 15 to 1363 = 465075

And, the total number of terms = 675

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 1363

= 465075/675 = 689

Thus, the average of the given odd numbers from 15 to 1363 = 689 Answer


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