Question : Find the average of odd numbers from 5 to 1013
Correct Answer 509
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 1013
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 1013 are
5, 7, 9, . . . . 1013
After observing the above list of the odd numbers from 5 to 1013 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 1013 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 1013
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 1013
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 1013
= 5 + 1013/2
= 1018/2 = 509
Thus, the average of the odd numbers from 5 to 1013 = 509 Answer
Method (2) to find the average of the odd numbers from 5 to 1013
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 1013 are
5, 7, 9, . . . . 1013
The odd numbers from 5 to 1013 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 1013
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 1013
1013 = 5 + (n – 1) × 2
⇒ 1013 = 5 + 2 n – 2
⇒ 1013 = 5 – 2 + 2 n
⇒ 1013 = 3 + 2 n
After transposing 3 to LHS
⇒ 1013 – 3 = 2 n
⇒ 1010 = 2 n
After rearranging the above expression
⇒ 2 n = 1010
After transposing 2 to RHS
⇒ n = 1010/2
⇒ n = 505
Thus, the number of terms of odd numbers from 5 to 1013 = 505
This means 1013 is the 505th term.
Finding the sum of the given odd numbers from 5 to 1013
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 1013
= 505/2 (5 + 1013)
= 505/2 × 1018
= 505 × 1018/2
= 514090/2 = 257045
Thus, the sum of all terms of the given odd numbers from 5 to 1013 = 257045
And, the total number of terms = 505
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 1013
= 257045/505 = 509
Thus, the average of the given odd numbers from 5 to 1013 = 509 Answer
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