Question

Find four geometric means between 4096 and 972.

an=a1*r^n-1
a6=972*r^6-1
4096=972r^5
4.2=r^5
1.3

972(1.3)=1296
1296(1.3)=1728
1728(1.3)=2304
2304(1.3)=3072

answer=1296,1728,2304,3072

Now that I look at it, do I have the a1 and a6 values mixed up? Because in the question 4096 is the first number and 972 was the last but I just put the smaller number first instead.

Answer 1

in your sequence term₁ = 4096
so a=4096 and
term₆ = ar^5 = 972

then r^5 = 972/4092
= 243/1024
= 3^5/4^5

r = 3/4

so the four geometric means would be term2, term3...
or 3072, 2304, 1728, and 1296

(notice if I multiply 1296 by 3/4 I get 972, which confirms my answer)

Don't turn the terms of a sequence around.
If │r│ > 1 the terms get larger, if │r│ < 1 the terms get smaller, which is our case.