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Question:     Find the average of odd numbers from 9 to 1349


Correct Answer  679

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 9 to 1349

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

The odd numbers from 9 to 1349 are

9, 11, 13, . . . . 1349

After observing the above list of the odd numbers from 9 to 1349 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 1349 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 9 to 1349

The First Term (a) = 9

The Common Difference (d) = 2

And the last term (ℓ) = 1349

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 9 to 1349

= 9 + 1349/2

= 1358/2 = 679

Thus, the average of the odd numbers from 9 to 1349 = 679 Answer

Method (2) to find the average of the odd numbers from 9 to 1349

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

The odd numbers from 9 to 1349 are

9, 11, 13, . . . . 1349

The odd numbers from 9 to 1349 form an Arithmetic Series in which

The First Term (a) = 9

The Common Difference (d) = 2

And the last term (ℓ) = 1349

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 9 to 1349

1349 = 9 + (n – 1) × 2

⇒ 1349 = 9 + 2 n – 2

⇒ 1349 = 9 – 2 + 2 n

⇒ 1349 = 7 + 2 n

After transposing 7 to LHS

⇒ 1349 – 7 = 2 n

⇒ 1342 = 2 n

After rearranging the above expression

⇒ 2 n = 1342

After transposing 2 to RHS

⇒ n = 1342/2

⇒ n = 671

Thus, the number of terms of odd numbers from 9 to 1349 = 671

This means 1349 is the 671th term.

Finding the sum of the given odd numbers from 9 to 1349

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 9 to 1349

= 671/2 (9 + 1349)

= 671/2 × 1358

= 671 × 1358/2

= 911218/2 = 455609

Thus, the sum of all terms of the given odd numbers from 9 to 1349 = 455609

And, the total number of terms = 671

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 9 to 1349

= 455609/671 = 679

Thus, the average of the given odd numbers from 9 to 1349 = 679 Answer


Similar Questions

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(2) Find the average of even numbers from 4 to 338

(3) Find the average of odd numbers from 13 to 25

(4) Find the average of odd numbers from 3 to 87

(5) Find the average of odd numbers from 15 to 191

(6) What is the average of the first 1483 even numbers?

(7) Find the average of the first 2190 even numbers.

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