Question:
Find the average of the first 717 odd numbers.
Correct Answer
717
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 717 odd numbers are
1, 3, 5, 7, 9, . . . . 717 th terms
Calculation of the sum of the first 717 odd numbers
We can find the sum of the first 717 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 717 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 717 odd number,
n = 717, a = 1, and d = 2
Thus, sum of the first 717 odd numbers
S717 = 717/2 [2 × 1 + (717 – 1) 2]
= 717/2 [2 + 716 × 2]
= 717/2 [2 + 1432]
= 717/2 × 1434
= 717/2 × 1434 717
= 717 × 717 = 514089
⇒ The sum of first 717 odd numbers (Sn) = 514089
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 717 odd numbers
= 7172 = 514089
⇒ The sum of first 717 odd numbers = 514089
Calculation of the Average of the first 717 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 717 odd numbers
= Sum of first 717 odd numbers/717
= 514089/717 = 717
Thus, the average of the first 717 odd numbers = 717 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 717 odd numbers = 717
Thus, the average of the first 717 odd numbers = 717 Answer
Similar Questions
(1) Find the average of the first 2028 even numbers.
(2) Find the average of even numbers from 10 to 1820
(3) Find the average of even numbers from 6 to 690
(4) Find the average of even numbers from 10 to 164
(5) Find the average of the first 2234 odd numbers.
(6) Find the average of even numbers from 10 to 1370
(7) Find the average of the first 460 odd numbers.
(8) Find the average of the first 2079 odd numbers.
(9) Find the average of the first 1617 odd numbers.
(10) Find the average of even numbers from 10 to 358