Question:
Find the average of odd numbers from 3 to 1227
Correct Answer
615
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 1227
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 1227 are
3, 5, 7, . . . . 1227
After observing the above list of the odd numbers from 3 to 1227 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 1227 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 1227
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1227
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 1227
= 3 + 1227/2
= 1230/2 = 615
Thus, the average of the odd numbers from 3 to 1227 = 615 Answer
Method (2) to find the average of the odd numbers from 3 to 1227
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 1227 are
3, 5, 7, . . . . 1227
The odd numbers from 3 to 1227 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1227
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 1227
1227 = 3 + (n – 1) × 2
⇒ 1227 = 3 + 2 n – 2
⇒ 1227 = 3 – 2 + 2 n
⇒ 1227 = 1 + 2 n
After transposing 1 to LHS
⇒ 1227 – 1 = 2 n
⇒ 1226 = 2 n
After rearranging the above expression
⇒ 2 n = 1226
After transposing 2 to RHS
⇒ n = 1226/2
⇒ n = 613
Thus, the number of terms of odd numbers from 3 to 1227 = 613
This means 1227 is the 613th term.
Finding the sum of the given odd numbers from 3 to 1227
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 1227
= 613/2 (3 + 1227)
= 613/2 × 1230
= 613 × 1230/2
= 753990/2 = 376995
Thus, the sum of all terms of the given odd numbers from 3 to 1227 = 376995
And, the total number of terms = 613
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 1227
= 376995/613 = 615
Thus, the average of the given odd numbers from 3 to 1227 = 615 Answer
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