Question : Find the average of the first 2154 odd numbers.
Correct Answer 2154
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 2154 odd numbers are
1, 3, 5, 7, 9, . . . . 2154 th terms
Calculation of the sum of the first 2154 odd numbers
We can find the sum of the first 2154 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 2154 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 2154 odd number,
n = 2154, a = 1, and d = 2
Thus, sum of the first 2154 odd numbers
S2154 = 2154/2 [2 × 1 + (2154 – 1) 2]
= 2154/2 [2 + 2153 × 2]
= 2154/2 [2 + 4306]
= 2154/2 × 4308
= 2154/2 × 4308 2154
= 2154 × 2154 = 4639716
⇒ The sum of first 2154 odd numbers (Sn) = 4639716
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 2154 odd numbers
= 21542 = 4639716
⇒ The sum of first 2154 odd numbers = 4639716
Calculation of the Average of the first 2154 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 2154 odd numbers
= Sum of first 2154 odd numbers/2154
= 4639716/2154 = 2154
Thus, the average of the first 2154 odd numbers = 2154 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 2154 odd numbers = 2154
Thus, the average of the first 2154 odd numbers = 2154 Answer
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