SOLUTION: The arithmetic means between the 6th and 9th term of an arithmetic sequence are 13.5 and 15 (in what order) what are the first and 20th terms of the sequence?

The arithmetic means between the 6th and 9th terms are the 
7th and 8th terms of the arithmetic sequence.  So we are 
asked to put the numbers in the two blanks below with 
question marks:

 ? ,___,___,___,___,___,13.5, 15,___,___,___,___,___,___,___,___,___,___,___, ? 

We know that the common difference d, is the difference between
any two consecutive terms.  We have the 7th and 8th terms, which
are consecutive, so the common difference = d = 15-13.5 = 1.5.

The formula for the nth term of an arithmetic sequence is:

an = a1 + (n - 1)d

Substitute n = 7 and d = 1.5

a7 = a1 + (7 - 1)(1.5)

Substitute a7 = 13.5 and simplify

13.5 = a1 + (6)(1.5)

13.5 = a1 + 9

 4.5 = a1    <-- the first term

Now we want to find the 20th term, so we use the formula

an = a1 + (n - 1)d

and substitute n = 20, d = 1.5 and  a1 = 4.5

a20 = 4.5 + (20 - 1)(1.5)

a20 = 4.5 + (19)(1.5)

a20 = 4.5 + 28.5

a20 = 33   <-- 20th term

Checking, here's the whole sequence through the 20th term:

4.5, 6,7.5, 9,10.5,12,13.5,15,16.5,18,19.5,21,22.5,24,25.5,27,28.5,30,31.5,33

Edwin