Sequence problem

[tug187]

i have no teacher to teach me this and i also have no textbook.. i have some booklets that explain parts but not as fully and as detailed as a textbook would.. im taking it correspondence with a family friend but he cant get a textbook for me yet.. I couldnt find the answer to this problem.. does any1 kno what to do to get started wit it?

A finite geometric sequence has t1 = 0.0124 and t2 = 0.256. How many terms does this sequence have if its middle term has the value of 156.25?

My family friend is on vacation for another week so i either have to wait that long to get help or maybe someone can help me now. thx

 

[Mrspi]

i have no teacher to teach me this and i also have no textbook.. i have some booklets that explain parts but not as fully and as detailed as a textbook would.. im taking it correspondence with a family friend but he cant get a textbook for me yet.. I couldnt find the answer to this problem.. does any1 kno what to do to get started wit it?

A finite geometric sequence has t1 = 0.0124 and t2 = 0.256. How many terms does this sequence have if its middle term has the value of 156.25?

My family friend is on vacation for another week so i either have to wait that long to get help or maybe someone can help me now. thx

"correspondence with a family friend?" I guess I've heard it all now.

Ok...

here are some ideas which might help you.

In a geometric sequence, each term is obtained by multiplying the previous term by the same number, known as the common ratio. To find the common ratio, divide a known term by the previous known term. In your example, divide t<SUB>2</SUB> by t<SUB>1</SUB>, or 0.256 by 0.0124. This quotient is r, the common ratio. I get

Any particular term t<SUB>n</SUB> = t<SUB>1</SUB>* r<SUP>(n-1)</SUP>

You know that the "middle" term is 156.25. Let's call this term t<SUB>n</SUB>.

So,
t<SUB>n</SUB> = t<SUB>1</SUB>*r<SUP>(n - 1)</SUP>

Substitute 156.25 for t<SUB>n</SUB>, 0.0124 for t<SUB>1</SUB>, and the value you got above for r.

Solve for n.....

And that will tell you which term is the middle term. From that, hopefully, you will be able to determine how many terms are in the sequence.

I wonder if you typed the problem correctly......I got an answer that is not a whole number, which is NOT possible in this kind of problem.

Please recheck your data.

 

[tug187]

I wonder if you typed the problem correctly......I got an answer that is not a whole number, which is NOT possible in this kind of problem.

Please recheck your data.

oops its 0.1024 not 0.0124

 

[Mrspi]

Ok...then use 0.1024 for t<SUB>1</SUB> and follow the steps I recommended. You should get an appropriate answer.

 

[tug187]

Ok...then use 0.1024 for t<SUB>1</SUB> and follow the steps I recommended. You should get an appropriate answer.

i got 156.25 = 0.1024*2.5^(n-1)

156.25 = .256^(n-1)

but i dont know how to bring the n down so i can solve for it
my first time doing a question where n has to be brought down and solved so i havnt learned it yet.

 

[Denis]

i got 156.25 = 0.1024*2.5^(n-1)

156.25 = .256^(n-1)

.1024 * 2.5^(n - 1) does not equal .256^(n - 1);
example: 3 * 2^2 = 3 * 4 = 12, not 6^2 = 36

Make your life easier and multiply your terms by 1000:
1: 1024, 2: 2560,......., n-1: 1562500

r = 2560 / 1024 = 2.5

1024 * 2.5^(n-1) = 1562500
2.5^(n-1) = 1562500 / 1024 [RULE: if a^x = b, then x = log(b) / log(a)]
n-1 = log(1562500 / 1024) / log(2.5)
n-1 = 8
n = 9 = middle term
So sequence has 17 terms.

If you can't follow that...well...

 

[soroban]

Hello, tug187!

A finite geometric sequence has: \(\displaystyle t_1\,=\,0.1024\) and \(\displaystyle t_2\,=\,0.256\)
How many terms does this sequence have if its middle term is \(\displaystyle 156.25\) ?

Let's start over . . .

The \(\displaystyle n^{th}\) term is: \(\displaystyle \:t_n\:=\:t_1\cdot r^{n-1}\)

We are told that: \(\displaystyle \,t_1\,=\,0.1024\)
. . and we know that: \(\displaystyle \,r \:=\:\frac{t_2}{t_1} \:=\:\frac{0.256}{0.1024} \:=\:2.5\)

Suppose there are \(\displaystyle N\) terms.
Since there is a "middle term", \(\displaystyle N\) must be an odd number,
. . and the middle term is the \(\displaystyle n^{th}\) term, where \(\displaystyle N\:=\:2n\,-\,1\)

Then the middle term is: \(\displaystyle \:t_n\:=\:0.1024(2.5)^{n-1} \:=\:156.25\)

Divide both sides by \(\displaystyle 0.1024:\;\;(2.5)^{n-1} \:=\:1525.878906\)

Take logs of both sides: \(\displaystyle \:\ln(2.5)^{n-1} \:=\:\ln(1525.878906)\)

. . and we have: \(\displaystyle \n-1)\cdot\ln(2.5) \:=\:\ln(1525.878906)\;\;\Rightarrow\;\;n\,-\,1\:=\:\frac{\ln(1525.878906)}{\ln(2.5)}\)

. . Then: \(\displaystyle \:n\,-\,1\:=\:8\;\;\Rightarrow\;\;n \:=\:9\)


Therefore: \(\displaystyle \,N\:=\:2(9)\,-\,1\:=\:17\)

. . The sequence has \(\displaystyle 17\) terms.