Acceptable input for a certain pocket calculator is a finite sequence of characters each of which is either a digit or a sign. The first character must be a digit, the last character must be a digit, and any character that is a sign must be followed by a digit. There are 10 possible digits and 4 possible signs.

link223mh 2022-10-16 Answered
Acceptable input for a certain pocket calculator is a finite sequence of characters each of which is either a digit or a sign. The first character must be a digit, the last character must be a digit, and any character that is a sign must be followed by a digit. There are 10 possible digits and 4 possible signs. If N k denotes the number of wuch acceptable sequences having length k, then N k is given recursively by
A . N 1 = 10 , N k = 10 N k 1
B . N 1 = 10 , N k = 14 N k 1
C . N 1 = 10 , N 2 = 100 , N k = 10 N k 1 + 40 N k 2
D . N 1 = 10 , N 2 = 140 , N k = 14 N k 1 + 40 N k 2
E . N 1 = 14 , N 2 = 196 , N k = 10 N k 1 + 14 N k 2
Regardless of the choices, how can one deduce the recursive formula?
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Answers (1)

Adalyn Pitts
Answered 2022-10-17 Author has 15 answers
A valid string of length k 1 either begins with valid string of length k 1 ( 10 N k 1 term comes from this) or has the ( k 1)th place filled with a sign,in which case the first k 2 symbols constitute a valid string of length k 1 (this gives 40 N k 2 as there 4 ways to choose the signs for k 1th place and 10 ways to choose a digit for kth place)
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