Question

can someone help me solve this problem?--
In order to make it easier to find the nth term of an geometric sequence given two terms of the sequence, several students decide to solve the formula for a geometric sequence for r. Which student has correctly done so?
images attached

Answer 1

well its just arithmetic, how would you approach it?

Answer 2

I am not really sure I have gone through notes and my teachers videos explaining it and I cant follow

Answer 3

does subtraction undo multiplicaton?

given:

y = ab, does y-a = ab - a ?

Answer 4

or rather,

does ab-a = b?

Answer 5

does 2(4) - 2 = 4?

Answer 6

yeah

Answer 7

y-a = ab - a yes, but thats not really what i was going for lol

does: ab - a = b? and for an example: does 2(4) - 2 = 4?

Answer 8

yes

Answer 9

prove it ..

Answer 10

how? 2+2=4 and 2x2=4

Answer 11

2(4) - 2 = 4

what is 2(4) - 2 ?

Answer 12

oh 8 lol sorry it late and im kinda tired

Answer 13

then in general: ab - a is not b

subtracing a1 from both sides is pointless.

Answer 14

2(4) - 2 = 8-2 = 6

and 6 is not 4

Answer 15

now, how do we undo a power?

Answer 16

umm divide?

Answer 17

no .. r^2, to undo it we root it

sqrt(r^2) = r^(2/2) = r

nrt(r^n) = r^(n/n) = r

Answer 18

ahh got it

Answer 19

good :)

Answer 20

hmm would it be the second one?

Answer 21

@amistre64

Answer 22

@Tommynaut

Answer 23

does the second one root it properly?

is r^[(n-1)/2] = r for all n?

Answer 24

no?

Answer 25

why is that a question?

is (n-1)/2 = 1 for all n, or can you find a value of n that makes it false?

Answer 26

ohh I that makes sense its hard for me to see it without the numbers

Answer 27

let n=11 ....

is (11-1)/2 = 1 ?

Answer 28

if not, then talking a sqrt is not an appropriate root

Answer 29

if numbers make it easier, then use numbers ..

10 = 5 r^(3)

what is r?

Answer 30

21 = 7 r^12

make life simpler for yourself ...

Answer 31

As @amistre64 has implied, we can rule out Nelly and Alison because their methods don't work at all - they try and subtract a1 from both sides but don't actually end up doing it.

If we look at the first two people's solutions, they both start with correctly dividing through both sides by a1. However, the next step is to cancel the n-1 which is the power of r.
To do that, we have to take the (n-1) root of both sides.
Cinthia just takes the square root of both sides (when there's no number on the root, it means it's the 2nd root, or the square root).
Antoinette DOES take the (n-1) root of both sides.

So who ends up with the right answer?

Answer 32

wouldn't r be the sequence? or the first number in the sequence

Answer 33

r is used since it is the first letter in the word "ratio"

r is a common ratio among the terms .... its not any term perse, its how we build the sequence.

Answer 34

it would be the first person. I didn't notice Cynthia only used one

Answer 35

a1 is the 1th term, the first term, the term of A that is at the start, first place.

Answer 36

That makes so much more sense. my teacher only explained it very briefly and didn't even mention that