SOLUTION: Insert arithmetic means between 7 and 71 in such a way that the 5th arithmetic mean is 27. Find the number of arithmetic means.

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Question 448025: Insert arithmetic means between 7 and 71 in such a way that the 5th arithmetic mean is 27. Find the number of arithmetic means.
Answer by Edwin McCravy(20056) About Me  (Show Source):
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Insert arithmetic means between 7 and 71 in such a way that the 5th arithmetic mean is 27. Find the number of arithmetic means.

7, __, __, __, __, 27, ..., 71

27 is the 5th term BEYOND 7, so it's the 6th term
INCLUDING 7.  So we begin with n=6

a1 =  7
a6 = 27

an = a1 + (n-1)d

Substitute n=6

a6 = a1 + (6-1)d
27 =  7 + 5d
20 = 5d
 4 = d

Now we substitute an = 71

an = a1 + (n-1)4
71 = 7 + (n-1)4
71 = 7 + 4(n-1)
71 = 7 + 4n - 4
71 = 3 + 4n
68 = 4n 
17 = n

So there are 17 terms, and the entire sequence is

7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71

Edwin