Question:
Find the average of odd numbers from 15 to 409
Correct Answer
212
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 409
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 409 are
15, 17, 19, . . . . 409
After observing the above list of the odd numbers from 15 to 409 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 409 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 409
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 409
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 15 to 409
= 15 + 409/2
= 424/2 = 212
Thus, the average of the odd numbers from 15 to 409 = 212 Answer
Method (2) to find the average of the odd numbers from 15 to 409
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 409 are
15, 17, 19, . . . . 409
The odd numbers from 15 to 409 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 409
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 15 to 409
409 = 15 + (n – 1) × 2
⇒ 409 = 15 + 2 n – 2
⇒ 409 = 15 – 2 + 2 n
⇒ 409 = 13 + 2 n
After transposing 13 to LHS
⇒ 409 – 13 = 2 n
⇒ 396 = 2 n
After rearranging the above expression
⇒ 2 n = 396
After transposing 2 to RHS
⇒ n = 396/2
⇒ n = 198
Thus, the number of terms of odd numbers from 15 to 409 = 198
This means 409 is the 198th term.
Finding the sum of the given odd numbers from 15 to 409
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 15 to 409
= 198/2 (15 + 409)
= 198/2 × 424
= 198 × 424/2
= 83952/2 = 41976
Thus, the sum of all terms of the given odd numbers from 15 to 409 = 41976
And, the total number of terms = 198
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 15 to 409
= 41976/198 = 212
Thus, the average of the given odd numbers from 15 to 409 = 212 Answer
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