Question:
Find the average of odd numbers from 3 to 523
Correct Answer
263
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 523
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 523 are
3, 5, 7, . . . . 523
After observing the above list of the odd numbers from 3 to 523 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 523 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 523
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 523
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 3 to 523
= 3 + 523/2
= 526/2 = 263
Thus, the average of the odd numbers from 3 to 523 = 263 Answer
Method (2) to find the average of the odd numbers from 3 to 523
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 523 are
3, 5, 7, . . . . 523
The odd numbers from 3 to 523 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 523
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 3 to 523
523 = 3 + (n – 1) × 2
⇒ 523 = 3 + 2 n – 2
⇒ 523 = 3 – 2 + 2 n
⇒ 523 = 1 + 2 n
After transposing 1 to LHS
⇒ 523 – 1 = 2 n
⇒ 522 = 2 n
After rearranging the above expression
⇒ 2 n = 522
After transposing 2 to RHS
⇒ n = 522/2
⇒ n = 261
Thus, the number of terms of odd numbers from 3 to 523 = 261
This means 523 is the 261th term.
Finding the sum of the given odd numbers from 3 to 523
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 3 to 523
= 261/2 (3 + 523)
= 261/2 × 526
= 261 × 526/2
= 137286/2 = 68643
Thus, the sum of all terms of the given odd numbers from 3 to 523 = 68643
And, the total number of terms = 261
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 3 to 523
= 68643/261 = 263
Thus, the average of the given odd numbers from 3 to 523 = 263 Answer
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