I'm doing homework and getting stuck at this problem
I'm confused with the concept of induction and I don't know how to do this
a) Prove by induction that 5^n + 2.3^(n+1) + 1 is divisible by 4 for all n ∈ N.
b) Prove that if 2^x = 5 then x is irrational.
any idea guys ?
Thanks,
Here is how I attempted the question
a) Prove by induction that 5^n + 2.3^(n+1) + 1 is divisible by 4 for all n ∈ N.
solution:
Let p(n) be the statement
then we have to prove p(n) is true for every natural number
p(0) = 5^0 + 2 . 3^1 + = 8 which is devisible by 4
p (1) = 5^1 + 2.3^2 + 1 = 24 which is again devisible by 4
so p(n) is true for every natural number
now I have to do the inductive hypothesis step which the most difficult bit in this question
what I have done is :
let K= N
p(K) = p (N) AND that implies p ( k+1) is true
5 ^ (k+1) + 2 . 3^(k+2) +1
I don't know what to do after this step
b) Prove that if 2^x = 5 then x is irrational.
i used two methods for this one
the first one is by calculating
2^x = 5
ln 2^x = ln 5
x ln 2 = ln 5
x = ln 5 / ln 2
x= 2.3219
so x is irrational
OR
by contradiction
suppose that x is rational
a,b are positive intergers
2^x = 5
2 ^ a/b = 5 raise both sides by the power (b)
2^ (a/b) ^b = 5 ^b
2 ^a = 5^b
and from that we can say x is irrational