Question

By successively using defining rule for \(M\left( n \right)\), find

(a)\(M\left( {102} \right)\) (b) \(M\left( {101} \right)\) (c) \(M\left( {99} \right)\)

(d) \(M\left( {97} \right)\) (e) \(M\left( {87} \right)\) (f) \(M\left( {76} \right)\)

Short answer

(a) \(M\left( {102} \right) = 92\)

(b) \(M\left( {101} \right) = 91\)

(c)\(M\left( {99} \right) = 91\)

(d) \(M\left( {97} \right) = 91\)

(e) \(M\left( {87} \right) = 91\)

(f) \(M\left( {76} \right) = 91\)

Step-by-step solution

To define the function

The McCarthy 91 function is defined using the rule:

\(M\left( n \right) = \left\{ \begin{aligned}{l}n - 10,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,n > 100\\M\left( {M\left( {n + 11} \right)} \right),\,\,\,\,\,if\,\,n \le 100\end{aligned} \right.\)

Step 2: (a) To find\(M\left( {102} \right)\)

The \(M\left( {102} \right)\) is given by,

\(\begin{aligned}{c}M\left( {102} \right) &= 102 - 10\\ &= 92\end{aligned}\)

\(\therefore M\left( {102} \right) = 92\)

(a) To find \(M\left( {102} \right)\)

The \(M\left( {102} \right)\) is given by,

\(\begin{aligned}{c}M\left( {102} \right) &= 102 - 10\\ &= 92\end{aligned}\)

\(\therefore M\left( {102} \right) = 92\)

(b) To find \(M\left( {101} \right)\)

The \(M\left( {101} \right)\) is given by,

\(\begin{aligned}{c}M\left( {101} \right) &= 101 - 10\\ &= 91\end{aligned}\)

\(\therefore M\left( {101} \right) = 91\)

(c) To find \(M\left( {99} \right)\)

The \(M\left( {99} \right)\) is given by,

Further, solve above expression,

\(\therefore M\left( {99} \right) = 91\)

(d) To find \(M\\(M\left( {97} \right)\)left( {97} \right)\)

The is given by,

Further, solve the above expression,

\(\therefore M\left( {97} \right) = 91\)

(e) To find \(M\left( {87} \right)\)

The \(M\left( {87} \right)\) is given by,

Further, solve the above expression,

Further, solve the above expression,

Further, solve the above expression,

Further, solve the above expression,

Further, solve the above expression,

Further, solve the above expression,

\(\therefore M\left( {87} \right) = 91\)

(f) To find \(M\left( {76} \right)\)

The \(M\left( {76} \right)\) is given by,

\(\therefore M\left( {76} \right) = 91\)