The first term of a geometric sequence is −1536 and the common ratio is 1/2 .

Question

Which term of the sequence is −96?

Answer 1

The formula for the n-th term in a geometric sequence is given by

 

a_n=ar^n-1 where a is the first term in the sequence, r is the common ratio, and a_n is the nth term.

 

In this case a = -1536, r = 1/2, and a_n=-96 is the nth term.  We just need to figure out what n is.

 

Rewriting the formula slightly, we know that a_n/a=r^n-1

 

Lets plug in what we know:

 

-96/-1536=(1/2)^n-1

 

This simplifies to (1/16)=(1/2)^n-1.  If you wanted to you could take the log base 1/2 of each side.  Or you can just ask what power do you have to raise 1/2 to in order to get 1/16.  If I multiply 1/2 by itself 4 times I get 1/16.  So the answer is 4 meaning that (1/16)=(1/2)⁴=(1/2)^n-1

 

This implies that 4=n-1 or

 

n=5.  So -96 is the 5th term in the sequence.

 

Good Luck

 

Cheers

 

-Chris