Dark Mode

857 views

0 votes

0

0

2 votes

NOTE

1,1,2,3,5,.......... is a fibonacci series.

Name the above fibonacci series as fibonacci 1.

Let us assume the n^{th} term of fibonacci 1 be f_{n}

then f_{n} = 1/√5 ( ( (1+√5)/2 )^{n }- ( ( 1-√5)/2 )^{n} ) (Consider this as a fact)

Now, come to our problem

Let us assume T(n) = a^{kn} b^{mn }........................................................... (1)

then T(n+1) = a^{kn+1} b^{mn+1}...............................................(2)

Now, it is given in the question that T(n+2) = T(n) * T(n+1)

=> T(n+2) = a^{kn} b^{mn} * a^{kn+1} b^{mn+1 }from equations 1 & 2

=> T (n+2) = a^{(kn + kn+1) } b^{(mn + mn+1)}

So, we are seeing that the power of 'a' in T(n+2) = summation of the powers of 'a' in T(n) and T(n+1)

Now, T (n+2) = a^{kn+2} b^{mn+2}

So, we get k_{n+2} = k_{n} +k_{n+1}

Thus the powers of 'a' forms a fibonacci series.

Similarly we can say it for the powers of 'b'

here m_{n+2} = m_{n} +m_{n+1}

Now, T(1) = a so, k_{1} = 1 and m_{1 }= 0

Again T(2) = b so, k_{2} = 0 and m_{2} = 1

Thus the powers of 'a' forms the fibonacci series 1,0,1,1,2,......

name the above fibonacci series as fibonacci 2

If we look at the series fibonacci 1 and fibonacci 2 carefully we get that the n^{th} term of fibonacci 2 is equal to (n-2)^{th} term of fibonacci 1 for n>=3.

So, the terms of fibonacci 2 are

k_{1 }= 1 k_{2} = 0 and k_{n }= f_{n-2 }for n>=3.

Similarly we get that the powers of 'b' forms the fibonacci series 0,1,1,2,.... name this series as Fibonacci 3

We get that n^{th} term of fibonacci 3 is equal to (n-1)^{th} term of Fibonacci 1

so, the terms of Fibonacci 3 are

m_{1} = 0, m_{2}=1 , m_{n} = f_{n-1} for n>=3

So, our required solution T(n) = a^{kn} b^{mn }= a^{fn-2}b^{fn-1 }for n>=3.

Ankit why are u keeping the strings intact as we keep in automata

Here we are given T(n) = T(n-1) * T(n-2)

Here, T(1) = a T(2) = b

T(3) = a*b = ab

T(4) = T(3) * T(2) = ab * b = ab^{2}

T(5) = T(4) * T(3) = ab^{2} * ab = a^{2}b^{3}

T(6) = T(5) * T(4) = a^{2}b^{3} * ab^{2 }= a^{3}b^{5}

........ and so on

Here we have to multiply a and b because the operator * (star) signifies it instead of concantenating it.

1