help please [induction] (1 Viewer)

[jason2kool]

prove is divisible by 7 for any positive integer n.

 

[Trebla]

prove is divisible by 7 for any positive integer n.

When n = 1
2³ⁿ - 1 = 7 which is divisible by 7
Assume the statement holds for n = k
i.e. 2^3k - 1 = 7M for some integer M
Required to prove the statement holds for n = k + 1
i.e. 2^{3k + 3} - 1 = 7N for some integer N
LHS = 2^{3k + 3} - 1
= 8(2^3k) - 1
= 8(7M + 1) - 1 by assumption
= 56M + 7
= 7(8M + 1)
= 7N where N = 8M + 1 as an integer
= RHS
The statement holds for n = k + 1 if the assumption holds. Since the statement is true for n = 1, then by induction it is true for all positive integers n.

 

[nightweaver066]

Step 1: n = 1, 2^3 - 1
= 8 - 1
= 7
True for n = 1

Step 2: Assume for n = k, 2^(3k) - 1 = 7m is true where m is a positive integer.
RTP for n = k + 1, 2^(3(k + 1)) - 1 = 7n where n is a positive integer
Proof: LHS: 2^(3(k+1)) - 1
= 2^(3k + 3) - 1
= 8 x 2^(3k) - 1
= 8(2^(3k) - 1) + 7
= 8(7m) + 7
= 7(8m + 1)
= 7n

True for n = k + 1 if true for n = k.

Step 3: Since it is true for n = 1 (step 1), it must also be true for n = 1 + 1 (step 2) and so on. Hence the result is generally true.

Edit: beaten by a minute..

 

[jason2kool]

where the fuck did the 8 come from? in step 3... thats where i get stuck

 

[jason2kool]

OHHH waitt, i get it..
thanks heaps guys

 

[nightweaver066]

where the fuck did the 8 come from? in step 3... thats where i get stuck

Indice laws.