Mathematical induction?

Question

Proof:
2^(4n) - 1 is divisable by 15

Answer 1

2^(4n) - 1 = 16^n -1
Proof by induction:
Show p(1) is true:
n=1, 16^1 -1=15 is divisible by 15

Assume p(n) is true:
assume 16^n-1 is divisible by 15

Prove p(n+1) is true:
16^(n+1) -1 = 16*16^n -1
= (15+1)*16^n -1
= 15*16^n + 16^n - 1
since 15*16^n is divisible by 15, and 16^n -1 is divisible by 15 (from the assumption), the whole thing is divisible by 15.

Therefore, p(n+1) is true.

Q.E.D.

Answer 2

Let the statement "2^(4n) - 1 is divisible by 15" be Pn.
When n=1, 2^(4n) - 1=16 - 1=15 which is divisible by 15.
Therefore, P1 is true.
Assume Pk is true for some positive integer k.
To show: P(k+1) is true.
Proof:
P(k+1)
=2^(4n+4) - 1
=(2^4)2^(4n) - 1
=16(2^(4n) - 1+1) - 1
=16(15m) +16 - 1 where m is an integer(since Pk is assumed to be true)
=16(15m) +15
=15(16m+1) which is divisible by 15
Therefore, P(k+1) is true if Pk is true. Also, P1 is true. Hence, by Mathematical Induction, Pn is true for all positive integer n.

Answer 3

1st step: Prove it for n=1.
2^(4*1) - 1 = 16 - 1 = 15
2nd step: Prove that if this expression is valid for n=m then it is also valid for n = m + 1
We assume this is true --->: 2^(4m)-1= 15*j .... (1) where j is an integer
Then it is required to prove that:
2^(4(m+1)) - 1 = 15*k where k is an integer that we must find in terms of j
Proof:
2^(4(m+1)) - 1 = 2^(4m + 4) - 1 = 2^(4m) * 2^(4) - 1 =
16*(2^4m) - 1
then subtract 1 and add 1 to the quantity between parentheses:
= 16* ((2^(4m)-1) + 1) - 1
then from (1) above 2^(4m) - 1 = 15*j where j is an integer then
=16*(15*j + 1) - 1 = 16*15*j + 16 - 1 = 16*15*j + 15
= 15*(16*j + 1)
clearly 16*j + 1 is an integer since j is an integer, let k = 16*j + 1 so that 2^(4(m+1)) - 1 = 15*k
This completes the proof.
Thank for reminding me of the good old days at school and good luck.

Answer 4

2^(4n) - 1 = 16^n - 1 = (15 + 1)^n - 1 = (...+...+....+...+1(afer n times)) -1 so 1 and -1 on cancel each other and the rest is divided by 15 because its powers of 15 multiplied by integers n and smthing else.
what is mathematical induction

Answer 5

2^(4n)-1=16*2^(n)-1 . And one can divide the entire 16*2^(n)-1 by 15 resulting in (16/15)*((2^(n))/15)-(1/15). that should get you somewhere. I do not care to fully explain such answers in case that are solutions to assigned school problems. Hope it helps