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Question:     Find the average of odd numbers from 3 to 831


Correct Answer  417

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 3 to 831

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

The odd numbers from 3 to 831 are

3, 5, 7, . . . . 831

After observing the above list of the odd numbers from 3 to 831 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 831 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 3 to 831

The First Term (a) = 3

The Common Difference (d) = 2

And the last term (ℓ) = 831

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 3 to 831

= 3 + 831/2

= 834/2 = 417

Thus, the average of the odd numbers from 3 to 831 = 417 Answer

Method (2) to find the average of the odd numbers from 3 to 831

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

The odd numbers from 3 to 831 are

3, 5, 7, . . . . 831

The odd numbers from 3 to 831 form an Arithmetic Series in which

The First Term (a) = 3

The Common Difference (d) = 2

And the last term (ℓ) = 831

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 3 to 831

831 = 3 + (n – 1) × 2

⇒ 831 = 3 + 2 n – 2

⇒ 831 = 3 – 2 + 2 n

⇒ 831 = 1 + 2 n

After transposing 1 to LHS

⇒ 831 – 1 = 2 n

⇒ 830 = 2 n

After rearranging the above expression

⇒ 2 n = 830

After transposing 2 to RHS

⇒ n = 830/2

⇒ n = 415

Thus, the number of terms of odd numbers from 3 to 831 = 415

This means 831 is the 415th term.

Finding the sum of the given odd numbers from 3 to 831

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 3 to 831

= 415/2 (3 + 831)

= 415/2 × 834

= 415 × 834/2

= 346110/2 = 173055

Thus, the sum of all terms of the given odd numbers from 3 to 831 = 173055

And, the total number of terms = 415

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 3 to 831

= 173055/415 = 417

Thus, the average of the given odd numbers from 3 to 831 = 417 Answer


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