1.

Using the digits 0,1,2,3,& 4, the number of ten digit sequences can be written so that the difference between any two consecutive digits is 1, is equal to k, then k/72 is equal to

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Solution :`A_(n)` denotes no. of `n` DIGITS sequence that end in 0 or 4
`B_(n)` denotes no. of `n` digit sequence that end in 1 or 3
`C_(n)` denotes of `n` digit sequence that end in 2
`A_(n+1)=B_(n)` ( `:'` each sequence in `A_(n+1)` can be converted into a sequence in `B_(n)` by DELETING its lat digit). `B_(n+1)=A_(n)+2C_(n)` ( `:'` each sequence in `A_(n)` can be converted into sequence in `B_(n+1)` by ADDING `a 1` (if it ENDS with 0) or a 3 ( ifit ends with 4) to its end 8 each sequence in`C_(n)` can be converted into a sequence in `B_(n+1)` by adding a 1 or 3 to it `C_(n+1)=B_(n)` (by deleting 2 at its end)
`B_(n+1)=3B_(n-1)` for `nge2, B_(1)=2, B_(2)=4`
`:. A_(10)+B_(10)+C_(10)=2B_(9)+B_(10)=4.3^(4)+4.3^(4)=648`


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