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Question:     Find the average of odd numbers from 5 to 975


Correct Answer  490

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 5 to 975

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

The odd numbers from 5 to 975 are

5, 7, 9, . . . . 975

After observing the above list of the odd numbers from 5 to 975 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 975 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 5 to 975

The First Term (a) = 5

The Common Difference (d) = 2

And the last term (ℓ) = 975

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 975

= 5 + 975/2

= 980/2 = 490

Thus, the average of the odd numbers from 5 to 975 = 490 Answer

Method (2) to find the average of the odd numbers from 5 to 975

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

The odd numbers from 5 to 975 are

5, 7, 9, . . . . 975

The odd numbers from 5 to 975 form an Arithmetic Series in which

The First Term (a) = 5

The Common Difference (d) = 2

And the last term (ℓ) = 975

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 975

975 = 5 + (n – 1) × 2

⇒ 975 = 5 + 2 n – 2

⇒ 975 = 5 – 2 + 2 n

⇒ 975 = 3 + 2 n

After transposing 3 to LHS

⇒ 975 – 3 = 2 n

⇒ 972 = 2 n

After rearranging the above expression

⇒ 2 n = 972

After transposing 2 to RHS

⇒ n = 972/2

⇒ n = 486

Thus, the number of terms of odd numbers from 5 to 975 = 486

This means 975 is the 486th term.

Finding the sum of the given odd numbers from 5 to 975

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 975

= 486/2 (5 + 975)

= 486/2 × 980

= 486 × 980/2

= 476280/2 = 238140

Thus, the sum of all terms of the given odd numbers from 5 to 975 = 238140

And, the total number of terms = 486

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 975

= 238140/486 = 490

Thus, the average of the given odd numbers from 5 to 975 = 490 Answer


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