Math Problem Help!!!

[stevensuarez513]

Hi I got this math problem that I have been trying to do for the past 3 hours however I can't seem to figure it out. I found one person who solved it but I can't figure out their working out. If you do solve it please show all the working out. Any help would be greatly appreciated

"Suppose n consecutive geometric terms are inserted between 1 and 2. Write the sum of these n terms, in terms of n."

Link to the other person who figured out the answer

 

[stapel]

"Suppose n consecutive geometric terms are inserted between 1 and 2. Write the sum of these n terms, in terms of n."

Is one to assume that the "n consecutive geometric terms" are to be inserted so that the endpoints, 1 and 2, "fit" in with the rest of the terms? I think this (unstated) assumption is necessary in order to find "the" answer. (Otherwise, there would be infinitely-many answers.)

I got this math problem that I have been trying to do for the past 3 hours however I can't seem to figure it out. I found one person who solved it but I can't figure out their working out....

Link to the other person who figured out the answer

...If you do solve it please show all the working out.

That person did "show all the working out". Please reply explaining clearly at which point you stop understanding the worked solution. Thank you!

 

[stevensuarez513]

Is one to assume that the "n consecutive geometric terms" are to be inserted so that the endpoints, 1 and 2, "fit" in with the rest of the terms? I think this (unstated) assumption is necessary in order to find "the" answer. (Otherwise, there would be infinitely-many answers.)


That person did "show all the working out". Please reply explaining clearly at which point you stop understanding the worked solution. Thank you!

Thanks for your reply. The part that I don't understand is how he/she found the ratio for the geometric series and expressed it in terms of n. I also didn't understand how they found the first term of the series and how they found that n+2nd term=2

 

[stapel]

The part that I don't understand is how he/she found the ratio for the geometric series and expressed it in terms of n.

For reference' sake, this is the reply posted at the other site:

1st term = 1
number of terms n+2
n+2nd term = 2

so 1(r)^(n+1) = 2 if r is ratio

so r = 2^(1/(n+1)

so sum of n+1 terms = (r^(n+1)-1)/(r-1) using sum formula
= (2 - 1)/(2^(1/(n+1))-1)
= 1/(2^(1/(n+1) - 1)

so sum of required terms = 1/(2^(1/(n+1) - 1) - 1

What is the definition of a term in a geometric sequence? What is the formula for the \(\displaystyle n\mbox{-th}\) term of a geometric sequence with first term \(\displaystyle a\) and common ratio \(\displaystyle r\)? Since there are \(\displaystyle n\) terms between the first (being 1) and the last (being 2), then what must be the form of the last term in terms of the formula for the \(\displaystyle n\mbox{-th}\) term?

Set up this equation, and then solve for the value of \(\displaystyle r.\)

I also didn't understand how they found the first term of the series and how they found that n+2nd term=2

You were given the first and last terms.

 

[stevensuarez513]

For reference' sake, this is the reply posted at the other site:


What is the definition of a term in a geometric sequence? What is the formula for the \(\displaystyle n\mbox{-th}\) term of a geometric sequence with first term \(\displaystyle a\) and common ratio \(\displaystyle r\)? Since there are \(\displaystyle n\) terms between the first (being 1) and the last (being 2), then what must be the form of the last term in terms of the formula for the \(\displaystyle n\mbox{-th}\) term?

Set up this equation, and then solve for the value of \(\displaystyle r.\)


You were given the first and last terms.

I understand know but im still unclear on how they know that n+2nd term=2. We dont know how many n terms there are and by the 2nd term do they mean 2

 

[stapel]

I understand know but im still unclear on how they know that n+2nd term=2.

Because they TOLD you that the last term was 2:

"Suppose n consecutive geometric terms are inserted between 1 and 2....

We dont know how many n terms there are...

And you don't need to know the value of \(\displaystyle n\), so this doesn't matter.

...and by the 2nd term do they mean 2

No. By the "\(\displaystyle n\, +\, 2\mbox{nd}\)" term (which is very non-standard English), they mean the "\(\displaystyle (n\, +\, 2)\mbox{-th}\)" term (pronounced as the "enn plus two-eth" term), being the last term of \(\displaystyle 1\) (the first term), followed by \(\displaystyle n\) (unknown) terms, followed by \(\displaystyle 2\) (the last term). This is \(\displaystyle n\, +\, 1\, +\, 1\, \)\(\displaystyle =\, n\, +\, 2\) terms. The last term, then, is the \(\displaystyle (n\, +\, 2)\mbox{-th}\) term. :wink: