Question:
Find the average of odd numbers from 5 to 419
Correct Answer
212
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 419
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 419 are
5, 7, 9, . . . . 419
After observing the above list of the odd numbers from 5 to 419 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 419 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 419
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 419
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 419
= 5 + 419/2
= 424/2 = 212
Thus, the average of the odd numbers from 5 to 419 = 212 Answer
Method (2) to find the average of the odd numbers from 5 to 419
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 419 are
5, 7, 9, . . . . 419
The odd numbers from 5 to 419 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 419
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 419
419 = 5 + (n – 1) × 2
⇒ 419 = 5 + 2 n – 2
⇒ 419 = 5 – 2 + 2 n
⇒ 419 = 3 + 2 n
After transposing 3 to LHS
⇒ 419 – 3 = 2 n
⇒ 416 = 2 n
After rearranging the above expression
⇒ 2 n = 416
After transposing 2 to RHS
⇒ n = 416/2
⇒ n = 208
Thus, the number of terms of odd numbers from 5 to 419 = 208
This means 419 is the 208th term.
Finding the sum of the given odd numbers from 5 to 419
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 419
= 208/2 (5 + 419)
= 208/2 × 424
= 208 × 424/2
= 88192/2 = 44096
Thus, the sum of all terms of the given odd numbers from 5 to 419 = 44096
And, the total number of terms = 208
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 419
= 44096/208 = 212
Thus, the average of the given odd numbers from 5 to 419 = 212 Answer
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