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Question:     Find the average of odd numbers from 13 to 1471


Correct Answer  742

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 13 to 1471

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

The odd numbers from 13 to 1471 are

13, 15, 17, . . . . 1471

After observing the above list of the odd numbers from 13 to 1471 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 1471 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 13 to 1471

The First Term (a) = 13

The Common Difference (d) = 2

And the last term (ℓ) = 1471

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 13 to 1471

= 13 + 1471/2

= 1484/2 = 742

Thus, the average of the odd numbers from 13 to 1471 = 742 Answer

Method (2) to find the average of the odd numbers from 13 to 1471

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

The odd numbers from 13 to 1471 are

13, 15, 17, . . . . 1471

The odd numbers from 13 to 1471 form an Arithmetic Series in which

The First Term (a) = 13

The Common Difference (d) = 2

And the last term (ℓ) = 1471

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 13 to 1471

1471 = 13 + (n – 1) × 2

⇒ 1471 = 13 + 2 n – 2

⇒ 1471 = 13 – 2 + 2 n

⇒ 1471 = 11 + 2 n

After transposing 11 to LHS

⇒ 1471 – 11 = 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 13 to 1471 = 730

This means 1471 is the 730th term.

Finding the sum of the given odd numbers from 13 to 1471

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 13 to 1471

= 730/2 (13 + 1471)

= 730/2 × 1484

= 730 × 1484/2

= 1083320/2 = 541660

Thus, the sum of all terms of the given odd numbers from 13 to 1471 = 541660

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 13 to 1471

= 541660/730 = 742

Thus, the average of the given odd numbers from 13 to 1471 = 742 Answer


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(4) Find the average of the first 3094 even numbers.

(5) Find the average of the first 3856 even numbers.

(6) Find the average of the first 3981 even numbers.

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