Question:
Find the average of odd numbers from 5 to 415
Correct Answer
210
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 415
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 415 are
5, 7, 9, . . . . 415
After observing the above list of the odd numbers from 5 to 415 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 415 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 415
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 415
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 415
= 5 + 415/2
= 420/2 = 210
Thus, the average of the odd numbers from 5 to 415 = 210 Answer
Method (2) to find the average of the odd numbers from 5 to 415
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 415 are
5, 7, 9, . . . . 415
The odd numbers from 5 to 415 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 415
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 415
415 = 5 + (n – 1) × 2
⇒ 415 = 5 + 2 n – 2
⇒ 415 = 5 – 2 + 2 n
⇒ 415 = 3 + 2 n
After transposing 3 to LHS
⇒ 415 – 3 = 2 n
⇒ 412 = 2 n
After rearranging the above expression
⇒ 2 n = 412
After transposing 2 to RHS
⇒ n = 412/2
⇒ n = 206
Thus, the number of terms of odd numbers from 5 to 415 = 206
This means 415 is the 206th term.
Finding the sum of the given odd numbers from 5 to 415
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 415
= 206/2 (5 + 415)
= 206/2 × 420
= 206 × 420/2
= 86520/2 = 43260
Thus, the sum of all terms of the given odd numbers from 5 to 415 = 43260
And, the total number of terms = 206
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 415
= 43260/206 = 210
Thus, the average of the given odd numbers from 5 to 415 = 210 Answer
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