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Question :    Find the average of odd numbers from 5 to 1429


Correct Answer  717

Solution & Explanation

Solution

Method (1) to find the average of the odd numbers from 5 to 1429

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

The odd numbers from 5 to 1429 are

5, 7, 9, . . . . 1429

After observing the above list of the odd numbers from 5 to 1429 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 1429 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 5 to 1429

The First Term (a) = 5

The Common Difference (d) = 2

And the last term (ℓ) = 1429

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 5 to 1429

= 5 + 1429/2

= 1434/2 = 717

Thus, the average of the odd numbers from 5 to 1429 = 717 Answer

Method (2) to find the average of the odd numbers from 5 to 1429

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

The odd numbers from 5 to 1429 are

5, 7, 9, . . . . 1429

The odd numbers from 5 to 1429 form an Arithmetic Series in which

The First Term (a) = 5

The Common Difference (d) = 2

And the last term (ℓ) = 1429

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 5 to 1429

1429 = 5 + (n – 1) × 2

⇒ 1429 = 5 + 2 n – 2

⇒ 1429 = 5 – 2 + 2 n

⇒ 1429 = 3 + 2 n

After transposing 3 to LHS

⇒ 1429 – 3 = 2 n

⇒ 1426 = 2 n

After rearranging the above expression

⇒ 2 n = 1426

After transposing 2 to RHS

⇒ n = 1426/2

⇒ n = 713

Thus, the number of terms of odd numbers from 5 to 1429 = 713

This means 1429 is the 713th term.

Finding the sum of the given odd numbers from 5 to 1429

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 5 to 1429

= 713/2 (5 + 1429)

= 713/2 × 1434

= 713 × 1434/2

= 1022442/2 = 511221

Thus, the sum of all terms of the given odd numbers from 5 to 1429 = 511221

And, the total number of terms = 713

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 5 to 1429

= 511221/713 = 717

Thus, the average of the given odd numbers from 5 to 1429 = 717 Answer


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