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Question :    Find the average of odd numbers from 9 to 243


Correct Answer  126

Solution & Explanation

Solution

Method (1) to find the average of the odd numbers from 9 to 243

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

The odd numbers from 9 to 243 are

9, 11, 13, . . . . 243

After observing the above list of the odd numbers from 9 to 243 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 243 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 9 to 243

The First Term (a) = 9

The Common Difference (d) = 2

And the last term (ℓ) = 243

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 243

= 9 + 243/2

= 252/2 = 126

Thus, the average of the odd numbers from 9 to 243 = 126 Answer

Method (2) to find the average of the odd numbers from 9 to 243

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

The odd numbers from 9 to 243 are

9, 11, 13, . . . . 243

The odd numbers from 9 to 243 form an Arithmetic Series in which

The First Term (a) = 9

The Common Difference (d) = 2

And the last term (ℓ) = 243

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 243

243 = 9 + (n – 1) × 2

⇒ 243 = 9 + 2 n – 2

⇒ 243 = 9 – 2 + 2 n

⇒ 243 = 7 + 2 n

After transposing 7 to LHS

⇒ 243 – 7 = 2 n

⇒ 236 = 2 n

After rearranging the above expression

⇒ 2 n = 236

After transposing 2 to RHS

⇒ n = 236/2

⇒ n = 118

Thus, the number of terms of odd numbers from 9 to 243 = 118

This means 243 is the 118th term.

Finding the sum of the given odd numbers from 9 to 243

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 243

= 118/2 (9 + 243)

= 118/2 × 252

= 118 × 252/2

= 29736/2 = 14868

Thus, the sum of all terms of the given odd numbers from 9 to 243 = 14868

And, the total number of terms = 118

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 243

= 14868/118 = 126

Thus, the average of the given odd numbers from 9 to 243 = 126 Answer


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