Question

List the first four terms of a geometric sequence with t1 = 4 and tn = -3tn-1.

Find the sixth term of a geometric sequence with t5 = 24 and t8 = 3

How would I solve for these two problems?

Answer 1

\[
\large t_1 = 4 \\ \large
t_n=-3t_{n-1} \\
\text{Put n = 2:} \\ \large
t_2 = -3t_1 = -3 * 4 = -12 \\ \large
\text{Put n = 3:} \large
\]

Answer 2

What do you get when you put n = 3?

Answer 3

You don't even have to use the formula anymore. It is a geometric sequence with -3 as the common ratio. So to get the third term, simply multiply the second term by -3. To get the fourth term, multiply the third term by -3.....etc.

Answer 4

I'll be honest, I'm completely lost. I understand how to find the sequence sums and sums for infinite geometric series but on this part of the assignment I just am mising something, I dont get it... Can you walk me through this with baby steps? I'm sure what I'm not getting is obvious and will make me feel like an idiot.

Answer 5

They have given you a formula to find any term in the sequence. The formula is:
\(\large t_n = -3 * t_{n-1}\). So the nth term is -3 times the (n-1)th term. That is, any term in the sequence is -3 times the previous term. Is this part clear?

Answer 6

\(t_1\) stands for the first term.
\(t_2\) stands for the second term.
\(t_3\) stands for the third term.
\(t_n\) stands for the nth term.
\(t_{n-1}\) stands for the (n-1)th term.

Answer 7

God I feel so stupid asking.....but what is the "nth" and "(n-1)th" term

Answer 8

It is a general case. If we say n = 7, then \(\large t_n\) becomes \(\large t_7\) and we are referring to the seventh term. So n is a variable, a general term, to which we can assign any positive integer value.

Answer 9

\(t_n\) stands for the nth term which is a general term.
If you assign a value to n then the general term will become specific.
If n = 1, then it becomes \(t_1\) which stands for the first term.

Answer 10

So would the first one be -12,36,-108,324?

Answer 11

Yes, the geometric sequence is: 4, -12, 36, -108, ....
They want only the first four terms.

Answer 12

Whoops, thanks for that catch. Also, is the second one just like the first one?

Answer 13

I mean the same way to find it, I know its a little different as in it only wants one term.

Answer 14

Wait I get it

Answer 15

24 is the 5th term in the sequence

Answer 16

and 3 is the 8th

Answer 17

The 6th term would be 16, correct?

Answer 18

Find the sixth term of a geometric sequence with t5 = 24 and t8 = 3

A geometric sequence has a common ratio. Let us call that 'r'. You take any term. multiply it by 'r' and you will get the next term.

They have given us the 5th term and the eighth term. We can figure out what the common ratio 'r' is:
t5 = 24
t6 = 24 * r
t7 = 24 * r * r = 24 * r^2
t8 = 24 * r^3 = 3
solve for r.

Answer 19

No wait

Answer 20

So was I wrong in thinking the 6th term is 16?

Answer 21

16 is incorrect.

Answer 22

Look thats the little stupid thing I was missing. For example t23 just signifies the 23rd term in the sequence.

Answer 23

Did you solve for r?

Answer 24

Oh wait gimme one second

Answer 25

For some reason I forgot about the 7th term, I just thought the 6th term would be right in the middle of the 5th and 8th

Answer 26

\[\large
t_5 = 24 \\ \large
t_6 = 24r \\ \large
t_7 = 24r^2 \\ \large
t_8 = 24r^3 = 3 \\ \large
r^3 = \frac{3}{24} = \frac 18 \\ \large
r = \frac 12 \\ \large
t_6 = 24r = 24 * \frac 12 = 12
\]