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Question:     Find the average of odd numbers from 3 to 1443


Correct Answer  723

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 3 to 1443

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

The odd numbers from 3 to 1443 are

3, 5, 7, . . . . 1443

After observing the above list of the odd numbers from 3 to 1443 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 1443 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 3 to 1443

The First Term (a) = 3

The Common Difference (d) = 2

And the last term (ℓ) = 1443

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 1443

= 3 + 1443/2

= 1446/2 = 723

Thus, the average of the odd numbers from 3 to 1443 = 723 Answer

Method (2) to find the average of the odd numbers from 3 to 1443

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

The odd numbers from 3 to 1443 are

3, 5, 7, . . . . 1443

The odd numbers from 3 to 1443 form an Arithmetic Series in which

The First Term (a) = 3

The Common Difference (d) = 2

And the last term (ℓ) = 1443

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 1443

1443 = 3 + (n – 1) × 2

⇒ 1443 = 3 + 2 n – 2

⇒ 1443 = 3 – 2 + 2 n

⇒ 1443 = 1 + 2 n

After transposing 1 to LHS

⇒ 1443 – 1 = 2 n

⇒ 1442 = 2 n

After rearranging the above expression

⇒ 2 n = 1442

After transposing 2 to RHS

⇒ n = 1442/2

⇒ n = 721

Thus, the number of terms of odd numbers from 3 to 1443 = 721

This means 1443 is the 721th term.

Finding the sum of the given odd numbers from 3 to 1443

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 1443

= 721/2 (3 + 1443)

= 721/2 × 1446

= 721 × 1446/2

= 1042566/2 = 521283

Thus, the sum of all terms of the given odd numbers from 3 to 1443 = 521283

And, the total number of terms = 721

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 1443

= 521283/721 = 723

Thus, the average of the given odd numbers from 3 to 1443 = 723 Answer


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(2) Find the average of the first 1813 odd numbers.

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(4) Find the average of even numbers from 10 to 1078

(5) Find the average of the first 3497 even numbers.

(6) Find the average of odd numbers from 5 to 849

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