Question

Insert 6 arithmetic means between 2 and 16. Also, prove that their sum is 6 times an arithmetic mean between 2 and 16.

Answer 1

To insert 6 arithmetic means between 2 and 16, you need to find the common difference, which is calculated by dividing the difference between the first and last term by the total number of terms. Then, the arithmetic means can be found by adding the common difference repeatedly. The sum of these means is 79, which is equal to 6 times the arithmetic mean of 9.

Step-by-step explanation:

To insert 6 arithmetic means between 2 and 16, we first need to find the common difference. The common difference can be calculated by dividing the difference between the first and last term by the total number of terms (including both the ends and the means):

Common difference = (16 - 2) / (6 + 2) = 14 / 8 = 1.75

Now we can find the arithmetic means by adding the common difference repeatedly:

1. 
   
2. 2
3. 
   
4. 2 + 1.75 = 3.75
5. 
   
6. 3.75 + 1.75 = 5.5
7. 
   
8. 5.5 + 1.75 = 7.25
9. 
   
10. 7.25 + 1.75 = 9
11. 
    
12. 9 + 1.75 = 10.75
13. 
    
14. 10.75 + 1.75 = 12.5
15. 
    
16. 12.5 + 1.75 = 14.25
17. 
    
18. 14.25 + 1.75 = 16

To prove that the sum of these arithmetic means is 6 times an arithmetic mean between 2 and 16, we can find the sum of the means and compare it to 6 times the arithmetic mean:

Sum of means = 3.75 + 5.5 + 7.25 + 9 + 10.75 + 12.5 + 14.25 + 16 = 79

Arithmetic mean between 2 and 16 = (2 + 16) / 2 = 18 / 2 = 9

6 times the arithmetic mean = 6 * 9 = 54

Since the sum of the means (79) is equal to 6 times the arithmetic mean (54), the statement is proven.