Question:
Find the average of the first 2097 odd numbers.
Correct Answer
2097
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 2097 odd numbers are
1, 3, 5, 7, 9, . . . . 2097 th terms
Calculation of the sum of the first 2097 odd numbers
We can find the sum of the first 2097 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 2097 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 2097 odd number,
n = 2097, a = 1, and d = 2
Thus, sum of the first 2097 odd numbers
S2097 = 2097/2 [2 × 1 + (2097 – 1) 2]
= 2097/2 [2 + 2096 × 2]
= 2097/2 [2 + 4192]
= 2097/2 × 4194
= 2097/2 × 4194 2097
= 2097 × 2097 = 4397409
⇒ The sum of first 2097 odd numbers (Sn) = 4397409
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 2097 odd numbers
= 20972 = 4397409
⇒ The sum of first 2097 odd numbers = 4397409
Calculation of the Average of the first 2097 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 2097 odd numbers
= Sum of first 2097 odd numbers/2097
= 4397409/2097 = 2097
Thus, the average of the first 2097 odd numbers = 2097 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 2097 odd numbers = 2097
Thus, the average of the first 2097 odd numbers = 2097 Answer
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