Question:
Find the average of odd numbers from 13 to 1041
Correct Answer
527
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 1041
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 1041 are
13, 15, 17, . . . . 1041
After observing the above list of the odd numbers from 13 to 1041 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 1041 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 1041
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 1041
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 1041
= 13 + 1041/2
= 1054/2 = 527
Thus, the average of the odd numbers from 13 to 1041 = 527 Answer
Method (2) to find the average of the odd numbers from 13 to 1041
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 1041 are
13, 15, 17, . . . . 1041
The odd numbers from 13 to 1041 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 1041
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 1041
1041 = 13 + (n – 1) × 2
⇒ 1041 = 13 + 2 n – 2
⇒ 1041 = 13 – 2 + 2 n
⇒ 1041 = 11 + 2 n
After transposing 11 to LHS
⇒ 1041 – 11 = 2 n
⇒ 1030 = 2 n
After rearranging the above expression
⇒ 2 n = 1030
After transposing 2 to RHS
⇒ n = 1030/2
⇒ n = 515
Thus, the number of terms of odd numbers from 13 to 1041 = 515
This means 1041 is the 515th term.
Finding the sum of the given odd numbers from 13 to 1041
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 1041
= 515/2 (13 + 1041)
= 515/2 × 1054
= 515 × 1054/2
= 542810/2 = 271405
Thus, the sum of all terms of the given odd numbers from 13 to 1041 = 271405
And, the total number of terms = 515
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 1041
= 271405/515 = 527
Thus, the average of the given odd numbers from 13 to 1041 = 527 Answer
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