Question:
Find the average of the first 1758 odd numbers.
Correct Answer
1758
Solution And 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 1758 odd numbers are
1, 3, 5, 7, 9, . . . . 1758 th terms
Calculation of the sum of the first 1758 odd numbers
We can find the sum of the first 1758 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 1758 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 1758 odd number,
n = 1758, a = 1, and d = 2
Thus, sum of the first 1758 odd numbers
S1758 = 1758/2 [2 × 1 + (1758 – 1) 2]
= 1758/2 [2 + 1757 × 2]
= 1758/2 [2 + 3514]
= 1758/2 × 3516
= 1758/2 × 3516 1758
= 1758 × 1758 = 3090564
⇒ The sum of first 1758 odd numbers (Sn) = 3090564
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 1758 odd numbers
= 17582 = 3090564
⇒ The sum of first 1758 odd numbers = 3090564
Calculation of the Average of the first 1758 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 1758 odd numbers
= Sum of first 1758 odd numbers/1758
= 3090564/1758 = 1758
Thus, the average of the first 1758 odd numbers = 1758 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 1758 odd numbers = 1758
Thus, the average of the first 1758 odd numbers = 1758 Answer
Similar Questions
(1) Find the average of the first 3892 odd numbers.
(2) Find the average of even numbers from 6 to 1398
(3) Find the average of odd numbers from 7 to 1033
(4) Find the average of the first 2297 even numbers.
(5) Find the average of the first 3514 odd numbers.
(6) Find the average of odd numbers from 11 to 133
(7) Find the average of odd numbers from 15 to 1001
(8) Find the average of the first 2374 odd numbers.
(9) Find the average of even numbers from 8 to 214
(10) Find the average of the first 2160 even numbers.