Question : Find the average of the first 1793 odd numbers.
Correct Answer 1793
Solution & Explanation
Explanation
Method to find the average
Step : (1) Find the sum of given numbers
Step: (2) Divide the sum of given number by the number of numbers. This will give the average of given numbers
The first 1793 odd numbers are
1, 3, 5, 7, 9, . . . . 1793 th terms
Calculation of the sum of the first 1793 odd numbers
We can find the sum of the first 1793 odd numbers by simply adding them, but this is a bit difficult. And if the list is long, it is very difficult to find their sum. So, in such a situation, we will use a formula to find the sum of given numbers that form a particular pattern.
Here, the list of the first 1793 odd numbers forms an Arithmetic series
In an Arithmetic Series, the common difference is the same. This means the difference between two consecutive terms are same in an Arithmetic Series.
The sum of n terms of an Arithmetic Series
Sn = n/2 [2a + (n – 1) d]
Where, n = number of terms, a = first term, and d = common difference
In the series of first 1793 odd number,
n = 1793, a = 1, and d = 2
Thus, sum of the first 1793 odd numbers
S1793 = 1793/2 [2 × 1 + (1793 – 1) 2]
= 1793/2 [2 + 1792 × 2]
= 1793/2 [2 + 3584]
= 1793/2 × 3586
= 1793/2 × 3586 1793
= 1793 × 1793 = 3214849
⇒ The sum of first 1793 odd numbers (Sn) = 3214849
Shortcut Method to find the sum of first n odd numbers
Thus, the sum of first n odd numbers = n2
Thus, the sum of first 1793 odd numbers
= 17932 = 3214849
⇒ The sum of first 1793 odd numbers = 3214849
Calculation of the Average of the first 1793 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 1793 odd numbers
= Sum of first 1793 odd numbers/1793
= 3214849/1793 = 1793
Thus, the average of the first 1793 odd numbers = 1793 Answer
Shortcut Trick to find the Average of the first n odd numbers
The average of the first 2 odd numbers
= 1 + 3/2
= 4/2 = 2
Thus, the average of the first 2 odd numbers = 2
The average of the first 3 odd numbers
= 1 + 3 + 5/3
= 9/3 = 3
Thus, the average of the first 3 odd numbers = 3
The average of the first 4 odd numbers
= 1 + 3 + 5 + 7/4
= 16/4 = 4
Thus, the average of the first 4 odd numbers = 4
The average of the first 5 odd numbers
= 1 + 3 + 5 + 7 + 9/5
= 25/5 = 5
Thus, the average of the first 5 odd numbers = 5
Thus, the Average of the the First n odd numbers = n
Thus, the average of the first 1793 odd numbers = 1793
Thus, the average of the first 1793 odd numbers = 1793 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 1107
(2) What is the average of the first 940 even numbers?
(3) Find the average of even numbers from 4 to 454
(4) Find the average of odd numbers from 5 to 1459
(5) What will be the average of the first 4908 odd numbers?
(6) Find the average of the first 3292 even numbers.
(7) Find the average of odd numbers from 15 to 523
(8) Find the average of the first 2908 even numbers.
(9) What is the average of the first 1729 even numbers?
(10) Find the average of odd numbers from 3 to 1013