Question

prove 8 is a factor of 9^n-1 with mathematical induction

Answer 1

do you know the steps involved in mathematical induction?

Answer 2

yeah i got the first one but i don't know how to do the second

Answer 3

so what was your first step?

Answer 4

prove for n=1

Answer 5

yes - and that is simple enough - correct?

Answer 6

next step is to ASSUME it is true for some value of n, say for n=k

Answer 7

then, using this assumption, you try to prove it is also true for n=k+1

Answer 8

understand?

Answer 9

but i dont have an actual equation

Answer 10

if something is a multiple of 8, then how can you represent that mathematically in an equation?

Answer 11

9^(k+1)-1=8k?

Answer 12

correct

Answer 13

/almost/

Answer 14

8(k+1)

Answer 15

first write the original statement as an equation.

you are told to prove that \(9^n-1\) is a multiple of 8, i.e you are told to prove:\[9^n-1=8m\]where n and m are integers

Answer 16

agreed?

Answer 17

sure

Answer 18

so you showed that for n=1 you have:\[9^1-1=9-1=8\]next you need to ASUME this equation is true for some n=k, i.e. ASSUME it is true to say that:\[9^k-1=8m\]finally, use this result to try to prove that \(9^{k+1}-1\) is also a multiple of 8.

Answer 19

so 9^(k+1)-1=8m

Answer 20

no - the 'm' here is not the same 'm' as in the result for the assumption

Answer 21

try to rewrite \(9^{k+1}-1\) in terms of \(9^k-1\) and things should start to make sense

Answer 22

HINT: \(9^{k+1}=9\times9^k\)

Answer 23

i'm lost

Answer 24

ok, I'll work through this one for you and hopefully you will see how to proceed with other similar problems

Answer 25

\[9^{k+1}-1=9\times9^k-1\]agreed?

Answer 26

sure

Answer 27

therefore:\[9^{k+1}-1=9\times9^k-1=9(9^k-1)+8\]agreed?

Answer 28

umm sure

Answer 29

not sure?

Answer 30

no

Answer 31

how about if I write it as:\[9^{k+1}-1=9\times9^k-1=9\times9^k-9+8=9(9^k-1)+8\]

Answer 32

its ok keep going

Answer 33

so next we use the result from the 2nd step of induction where we ASSUMED that:\[9^k-1=8m\]to get:\[9^{k+1}-1=9\times9^k-1=9(9^k-1)+8=9(8m)+8\]make sense so far?

Answer 34

u lost me after ''to get"

Answer 35

:)

Answer 36

ok, let me try again...

Answer 37

I'm just going to connect my writing pad to make it easier to explain

Answer 38

ok