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Question:     Find the average of odd numbers from 13 to 1039


Correct Answer  526

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 13 to 1039

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

The odd numbers from 13 to 1039 are

13, 15, 17, . . . . 1039

After observing the above list of the odd numbers from 13 to 1039 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 1039 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 13 to 1039

The First Term (a) = 13

The Common Difference (d) = 2

And the last term (ℓ) = 1039

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 13 to 1039

= 13 + 1039/2

= 1052/2 = 526

Thus, the average of the odd numbers from 13 to 1039 = 526 Answer

Method (2) to find the average of the odd numbers from 13 to 1039

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

The odd numbers from 13 to 1039 are

13, 15, 17, . . . . 1039

The odd numbers from 13 to 1039 form an Arithmetic Series in which

The First Term (a) = 13

The Common Difference (d) = 2

And the last term (ℓ) = 1039

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 13 to 1039

1039 = 13 + (n – 1) × 2

⇒ 1039 = 13 + 2 n – 2

⇒ 1039 = 13 – 2 + 2 n

⇒ 1039 = 11 + 2 n

After transposing 11 to LHS

⇒ 1039 – 11 = 2 n

⇒ 1028 = 2 n

After rearranging the above expression

⇒ 2 n = 1028

After transposing 2 to RHS

⇒ n = 1028/2

⇒ n = 514

Thus, the number of terms of odd numbers from 13 to 1039 = 514

This means 1039 is the 514th term.

Finding the sum of the given odd numbers from 13 to 1039

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 13 to 1039

= 514/2 (13 + 1039)

= 514/2 × 1052

= 514 × 1052/2

= 540728/2 = 270364

Thus, the sum of all terms of the given odd numbers from 13 to 1039 = 270364

And, the total number of terms = 514

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 13 to 1039

= 270364/514 = 526

Thus, the average of the given odd numbers from 13 to 1039 = 526 Answer


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