Question:
Find the average of odd numbers from 13 to 409
Correct Answer
211
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 409
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 409 are
13, 15, 17, . . . . 409
After observing the above list of the odd numbers from 13 to 409 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 409 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 409
The First Term (a) = 13
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 13 to 409
= 13 + 409/2
= 422/2 = 211
Thus, the average of the odd numbers from 13 to 409 = 211 Answer
Method (2) to find the average of the odd numbers from 13 to 409
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 409 are
13, 15, 17, . . . . 409
The odd numbers from 13 to 409 form an Arithmetic Series in which
The First Term (a) = 13
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 13 to 409
409 = 13 + (n – 1) × 2
⇒ 409 = 13 + 2 n – 2
⇒ 409 = 13 – 2 + 2 n
⇒ 409 = 11 + 2 n
After transposing 11 to LHS
⇒ 409 – 11 = 2 n
⇒ 398 = 2 n
After rearranging the above expression
⇒ 2 n = 398
After transposing 2 to RHS
⇒ n = 398/2
⇒ n = 199
Thus, the number of terms of odd numbers from 13 to 409 = 199
This means 409 is the 199th term.
Finding the sum of the given odd numbers from 13 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 13 to 409
= 199/2 (13 + 409)
= 199/2 × 422
= 199 × 422/2
= 83978/2 = 41989
Thus, the sum of all terms of the given odd numbers from 13 to 409 = 41989
And, the total number of terms = 199
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 409
= 41989/199 = 211
Thus, the average of the given odd numbers from 13 to 409 = 211 Answer
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