Question

Let b be a fixed integer and j a fixed positive integer. Show that if$\mathbf{P}\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{,}\mathbf{P}\mathbf{(}\mathbf{b}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{P}\mathbf{(}\mathbf{b}\mathbf{+}\mathbf{j}\mathbf{)}$ are true and$\left[P\left(b{)}^{\wedge }P\right(b+1{)}^{\wedge }{\dots }^{\wedge }P\left(k\right)\right]\mathbf{\to }\mathbf{P}\mathbf{(}\mathbf{k}\mathbf{+}\mathbf{1}\mathbf{)}$ is true for every integer $\mathit{k}\mathbf{\ge }\mathit{b}\mathbf{+}\mathit{j}$, then$\mathbf{P}\mathbf{\left(}\mathbf{n}\mathbf{\right)}$ is true for all integersn with$\mathit{n}\mathbf{\ge }\mathit{b}$

Short answer

The$P\left(n\right)$ holds true for all the integersn with $n\ge b$.

Step-by-step solution

Identification of given data

The given data can be listed below as:

• The fixed integer isb .
• The fixed positive integer isj .

Significance of the strong induction

The strong induction is described as a technique different from the simple induction in order to prove a particular theorem and statement. The strong induction takes less time compared to the simple induction while proving a theorem or statement.

Determination of the proof of P(n)

In the basic step, when$n=b,b+1,...,b+j$, it is given that$P\left(b\right),P(b+1),...,P(b+j)$holds true.

In the inductive step, it has been assumed that $P\left(b\right),P(b+1),...,P(b+j)$holds true by$k\ge b+j$ . It is needed to be proved that$P(k+1)$ holds true.

As$P\left(b\right),P(b+1),...,P(b+j)$ holds true, then$P\left(b{)}^{\wedge }P\right(b+1{)}^{\wedge }{\dots }^{\wedge }P\left(k\right)$ also holds true. As$P\left(b{)}^{\wedge }P\right(b+1{)}^{\wedge }\dots {\wedge }^{\wedge }P\left(k\right)$ holds true and also$\left[P\left(b\right)\wedge P(b+1{)}^{\wedge }{\dots }^{\wedge }P(k)\right]\to P(k+1)$ holds true. Hence,$P(k+1)$ holds true.

Thus,$P\left(n\right)$ holds true for all the integersn with$n\ge b$ .