Question:
Find the average of the first 943 odd numbers.
Correct Answer
943
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 943 odd numbers are
1, 3, 5, 7, 9, . . . . 943 th terms
Calculation of the sum of the first 943 odd numbers
We can find the sum of the first 943 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 943 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 943 odd number,
n = 943, a = 1, and d = 2
Thus, sum of the first 943 odd numbers
S943 = 943/2 [2 × 1 + (943 – 1) 2]
= 943/2 [2 + 942 × 2]
= 943/2 [2 + 1884]
= 943/2 × 1886
= 943/2 × 1886 943
= 943 × 943 = 889249
⇒ The sum of first 943 odd numbers (Sn) = 889249
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 943 odd numbers
= 9432 = 889249
⇒ The sum of first 943 odd numbers = 889249
Calculation of the Average of the first 943 odd numbers
Formula to find the Average
Average = Sum of given numbers/Number of numbers
Thus, The average of the first 943 odd numbers
= Sum of first 943 odd numbers/943
= 889249/943 = 943
Thus, the average of the first 943 odd numbers = 943 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 943 odd numbers = 943
Thus, the average of the first 943 odd numbers = 943 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1502
(2) Find the average of odd numbers from 13 to 193
(3) Find the average of the first 4839 even numbers.
(4) Find the average of the first 2036 even numbers.
(5) Find the average of even numbers from 10 to 972
(6) Find the average of odd numbers from 13 to 1289
(7) What is the average of the first 343 even numbers?
(8) Find the average of the first 2518 even numbers.
(9) Find the average of even numbers from 12 to 1102
(10) Find the average of even numbers from 4 to 570