Geometric Sequence: T1=0.1024, T2=0.256, Middle Term=156.25

[ThomasMagnus]

A finite geometric sequence has t1 = 0.1024 and t2 = 0.256. How many terms does this sequence have if its middle term has a value of 156.25?

My Solution

Common Ratio: T2/T1=(.256)/(.1024)=2.5
What term # is the middle term?
tn=ar^n-1
a=0.1024
r=2.5
tn=156.25

(156.25)=(0.1024)(2.5)^n-1

1525=(2.5)^n-1

[Log(1525)/Log(2.5)]+1=n
n=8+1=9

n is the middle term so final term should be 2n. 18 terms in the sequence

From the way I have seen other people do this question, they get the answer "17 terms". Why am I 1 term off? Can you help me with what I am doing wrong. Or am I doing it correctly :)

Thanks!

 

[eumyang]

n is the middle term so final term should be 2n. 18 terms in the sequence

Not so! Can a sequence with an even number of terms have a middle term?

For example, if I have a simple sequence of natural numbers where 3 is the 3rd and middle term:
1 2 3 ...
You're saying that there are 2n, or 6 terms in the sequence:
1 2 3 4 5 6
See the problem? If 3 is the middle term, then there are 2n - 1, or 5 terms in the sequence:
1 2 3 4 5

So there should be 17 terms in the sequence in your problem.

 

[ThomasMagnus]

Not so! Can a sequence with an even number of terms have a middle term?

For example, if I have a simple sequence of natural numbers where 3 is the 3rd and middle term:
1 2 3 ...
You're saying that there are 2n, or 6 terms in the sequence:
1 2 3 4 5 6
See the problem? If 3 is the middle term, then there are 2n - 1, or 5 terms in the sequence:
1 2 3 4 5

So there should be 17 terms in the sequence in your problem.

I'm struggling to understand what you are saying.

"You're saying that there are 2n, or 6 terms in the sequence:
1 2 3 4 5 6"

Isn't there six terms in this sequence?Thanks :)

 

[eumyang]

I'm struggling to understand what you are saying.

"You're saying that there are 2n, or 6 terms in the sequence:
1 2 3 4 5 6"

Isn't there six terms in this sequence?

Yes, but I consider this sequence to have no "middle term", because there are an even number of terms. If you have a 5 term sequence (2n - 1):
1 2 3 4 5
... then 3 would be the middle term.

 

[ThomasMagnus]

So the middle term is always defined as 2n-1?

 

[eumyang]

Yes, if there are n terms in a sequence and n is odd.

 

[HallsofIvy]

So the middle term is always defined as 2n-1?

Yes, if there are n terms in a sequence and n is odd.

No, if there are n terms, for n larger than 1, then 2n- 1 is larger than n! If n is odd, then the middle term is indexed by (n+ 1)/2.

 

[eumyang]

No, if there are n terms, for n larger than 1, then 2n- 1 is larger than n! If n is odd, then the middle term is indexed by (n+ 1)/2.

Wow. I need to wake up. I think I meant to say: if the middle term of a sequence is the nth term, then there are 2n - 1 terms in the sequence.