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MCQs Math


Question:     Find the average of odd numbers from 13 to 341


Correct Answer  177

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 13 to 341

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

The odd numbers from 13 to 341 are

13, 15, 17, . . . . 341

After observing the above list of the odd numbers from 13 to 341 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 341 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 13 to 341

The First Term (a) = 13

The Common Difference (d) = 2

And the last term (ℓ) = 341

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 341

= 13 + 341/2

= 354/2 = 177

Thus, the average of the odd numbers from 13 to 341 = 177 Answer

Method (2) to find the average of the odd numbers from 13 to 341

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

The odd numbers from 13 to 341 are

13, 15, 17, . . . . 341

The odd numbers from 13 to 341 form an Arithmetic Series in which

The First Term (a) = 13

The Common Difference (d) = 2

And the last term (ℓ) = 341

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 341

341 = 13 + (n – 1) × 2

⇒ 341 = 13 + 2 n – 2

⇒ 341 = 13 – 2 + 2 n

⇒ 341 = 11 + 2 n

After transposing 11 to LHS

⇒ 341 – 11 = 2 n

⇒ 330 = 2 n

After rearranging the above expression

⇒ 2 n = 330

After transposing 2 to RHS

⇒ n = 330/2

⇒ n = 165

Thus, the number of terms of odd numbers from 13 to 341 = 165

This means 341 is the 165th term.

Finding the sum of the given odd numbers from 13 to 341

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 341

= 165/2 (13 + 341)

= 165/2 × 354

= 165 × 354/2

= 58410/2 = 29205

Thus, the sum of all terms of the given odd numbers from 13 to 341 = 29205

And, the total number of terms = 165

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 341

= 29205/165 = 177

Thus, the average of the given odd numbers from 13 to 341 = 177 Answer


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(4) Find the average of odd numbers from 13 to 173

(5) Find the average of even numbers from 4 to 1330

(6) Find the average of the first 3238 odd numbers.

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