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Question:     Find the average of odd numbers from 5 to 813


Correct Answer  409

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 5 to 813

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

The odd numbers from 5 to 813 are

5, 7, 9, . . . . 813

After observing the above list of the odd numbers from 5 to 813 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 813 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 5 to 813

The First Term (a) = 5

The Common Difference (d) = 2

And the last term (ℓ) = 813

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 5 to 813

= 5 + 813/2

= 818/2 = 409

Thus, the average of the odd numbers from 5 to 813 = 409 Answer

Method (2) to find the average of the odd numbers from 5 to 813

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

The odd numbers from 5 to 813 are

5, 7, 9, . . . . 813

The odd numbers from 5 to 813 form an Arithmetic Series in which

The First Term (a) = 5

The Common Difference (d) = 2

And the last term (ℓ) = 813

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 5 to 813

813 = 5 + (n – 1) × 2

⇒ 813 = 5 + 2 n – 2

⇒ 813 = 5 – 2 + 2 n

⇒ 813 = 3 + 2 n

After transposing 3 to LHS

⇒ 813 – 3 = 2 n

⇒ 810 = 2 n

After rearranging the above expression

⇒ 2 n = 810

After transposing 2 to RHS

⇒ n = 810/2

⇒ n = 405

Thus, the number of terms of odd numbers from 5 to 813 = 405

This means 813 is the 405th term.

Finding the sum of the given odd numbers from 5 to 813

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 5 to 813

= 405/2 (5 + 813)

= 405/2 × 818

= 405 × 818/2

= 331290/2 = 165645

Thus, the sum of all terms of the given odd numbers from 5 to 813 = 165645

And, the total number of terms = 405

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 5 to 813

= 165645/405 = 409

Thus, the average of the given odd numbers from 5 to 813 = 409 Answer


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