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Let f(n) denotes the number of different...

Let f(n) denotes the number of different ways, the positive integer n ca be expressed as the sum of the 1's and 2's. for example, f(4)=5.
i.e., `4=1+1+1+1`
`=1+1+2=1+2+1=2+1+1=2+2`
Q. The number of solutions of the equation `f(n)=n`, where `n in N` is

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
C
`becausef(1)=.^(1)C_(0)=1,f(2)=.^(2)C_(0)+.^(1)C_(1)=2,f(3)=.^(3)C_(0)+.^(2)C_(1)=3`,
f(4)=5
and `f(5)=.^(5)C_(0)+.^(4)C_(1)+.^(3)C_(2)=8`
thus, we say that `f(x) gt n` for n=4,5,6, . .
Hence, number of solutions for f(n)=5=n is 3.
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