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Question:     Find the average of odd numbers from 5 to 919


Correct Answer  462

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 5 to 919

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

The odd numbers from 5 to 919 are

5, 7, 9, . . . . 919

After observing the above list of the odd numbers from 5 to 919 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 919 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 5 to 919

The First Term (a) = 5

The Common Difference (d) = 2

And the last term (ℓ) = 919

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 919

= 5 + 919/2

= 924/2 = 462

Thus, the average of the odd numbers from 5 to 919 = 462 Answer

Method (2) to find the average of the odd numbers from 5 to 919

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

The odd numbers from 5 to 919 are

5, 7, 9, . . . . 919

The odd numbers from 5 to 919 form an Arithmetic Series in which

The First Term (a) = 5

The Common Difference (d) = 2

And the last term (ℓ) = 919

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 919

919 = 5 + (n – 1) × 2

⇒ 919 = 5 + 2 n – 2

⇒ 919 = 5 – 2 + 2 n

⇒ 919 = 3 + 2 n

After transposing 3 to LHS

⇒ 919 – 3 = 2 n

⇒ 916 = 2 n

After rearranging the above expression

⇒ 2 n = 916

After transposing 2 to RHS

⇒ n = 916/2

⇒ n = 458

Thus, the number of terms of odd numbers from 5 to 919 = 458

This means 919 is the 458th term.

Finding the sum of the given odd numbers from 5 to 919

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 919

= 458/2 (5 + 919)

= 458/2 × 924

= 458 × 924/2

= 423192/2 = 211596

Thus, the sum of all terms of the given odd numbers from 5 to 919 = 211596

And, the total number of terms = 458

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 919

= 211596/458 = 462

Thus, the average of the given odd numbers from 5 to 919 = 462 Answer


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