How do you write an equation for the nth term of geometric sequence -6561, 1458, -324?

1 Answer
Mar 8, 2016

Answer:

#color(blue)((-1)^n ar^(n-1)) #

Explanation:

The general structure of a geometric sequence is:

#ar^0" , "ar^1" , "ar^2" , "ar^3"....."ar^n.#

So lets just look at the numbers for the moment

From the above it is quite evident that you can find #r# by applying:

#" "(ar^1)/(ar^0) = r#

So we have

#"Term (n) a " r^(n-1)" "ar^(n-1)#

#" 1 -6561 0 -6561"#

#" 2 -6561 r +1458"#

#" 3 -6561 "r^2" -324"#

Ignoring the signs to start with. We can allow for that later.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#" "(ar^1)/(ar^0) = 1458/6561 = 2/9 = r#

Test: The next number should be of magnitude 324

#" "6561 xx (2/9)^2 = 324# as required
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider the alternating positive negative

If we use #(-1)^n# this will give us the correct sign

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Final structure")#

#color(blue)((-1)^n ar^(n-1)) #