Question

use mathematical induction to show that (7^n)-1 is divisible by 6

Answer 1

Let P(n) be the statement 7^n - 1 is divisible by 6.

Then first of all P(1) is clearly true as 7^1 - 1 = 6.

Now suppose that P(k) is true for some k >= 1, we want to show that we can deduce that P(k+1) is true.

Answer 2

Well, P(k) being true means that 7^k - 1 is divisible by 6

What does that imply for 7^(k+1) - 1

The 'trick' here is finding a way to express this last expression in terms of 7^k - 1.

Think about it for a few minutes.

Answer 3

so for n=1, it is obiviously true since 6|6

now assume for some k that \[6|(7^k-1)=>7^k-1 =6a, a \in \mathbb{Z} \]
so now we to need to show the expression is true for k+1
\[7^{k+1}-1=7^k7-1=7(7^k-\frac{1}{7})=7(7^k-1+\frac{6}{7})=7(7^k-1)+7(\frac{6}{7})\]
\[7(6a)+6=6(7a+1)\]

Answer 4

but since a is an integer then 7a+1 is an integer
so we have that \[6|(7^k-1)\] for all integer k>=1

Answer 5

so james u mean
the thesis will b
7^(k+1)-1

Answer 6

I'm not exactly sure how you're using the word thesis here.

We are trying to show that P(k) ==> P(k+1). I.e., we can deduce the statement P(k+1) from P(k).

Answer 7

myininaya, where does "a" come from? We started with variable n

Answer 8

a is integer that i pulled out of the skies

Answer 9

ok i will tell you again
if \[6|(7^k-1) => 6a=7^k-1, a \in \mathbb{Z}\]

Answer 10

We can also do this in an "a"-less way.

Note that
\[7^{k+1} = 7.7^k - 1 = 7(7^k -1) + 6\]

Now we already know something about that expression in brackets,
7^k - 1 ....

Answer 11

6|something means there is some integer k such that 6k=something

Answer 12

i usually say k but something else was called k so i used a

Answer 13

James, for less confusion, write:

\[7 \dot\ 7^k -1\]

Answer 14

i have actually seen 7.7^k written in a paper and it was defined as multiplication

Answer 15

well not that exact expression

Answer 16

but . was defined as multiplication

Answer 17

we can make any symbol to define any operation we want

Answer 18

Myininaya, stop defending James

Answer 19

lol

Answer 20

lol

Answer 21

i like james

Answer 22

ok james its that all

Answer 23

I know you do

Answer 24

You're a biological computer. Biological computers are not allowed to have "favorites" or take sides...:

:P

Answer 25

thanks ...
@remainder: do you have your solution?

Answer 26

thanks every1