Chapter 11.5, Problem 85E

(a)

To determine

To prove: The given binomial theorem for $n=1$.

(b)

To determine

The statement for which one can assumed have statement is true and the statement which has to be proven. If the binomial expression is

${\left(a+b\right)}^n=\left(\begin{array}{l}n\\ 0\end{array}\right)a^n+\left(\begin{array}{l}n\\ 1\end{array}\right)a^{n-1}b+\left(\begin{array}{l}n\\ 2\end{array}\right)a^{n-2}{b}^2+\cdots +\left(\begin{array}{c}n\\ n-1\end{array}\right)a{b}^{n-1}+\left(\begin{array}{l}n\\ n\end{array}\right)b^n$

Replace $n=k$ and $n=k+1$.

(c)

To determine

To calculate: The statement for $n=k$ that is assumed true to which $\left(a+b\right)$ is multiplied and simplify.

(d)

To determine

To calculate: Collecting like terms on the right-hand side of the last statement, at what time one has

$\begin{array}{c}{\left(a+b\right)}^{k+1}=\left(\begin{array}{l}k\\ 0\end{array}\right)a^{k+1}+\left(\begin{array}{l}k\\ 0\end{array}\right)a^{k}b+\left(\begin{array}{l}k\\ 1\end{array}\right)a^{k}b+\left(\begin{array}{l}k\\ 1\end{array}\right)a^{k-1}{b}^2+\left(\begin{array}{l}k\\ 2\end{array}\right)a^{k-1}{b}^2+\left(\begin{array}{l}k\\ 2\end{array}\right)a^{k-2}{b}^3\\ +\cdots +\left(\begin{array}{c}k\\ k-1\end{array}\right)a^2{b}^{k-1}+\left(\begin{array}{c}k\\ k-1\end{array}\right)a{b}^k+\left(\begin{array}{l}k\\ k\end{array}\right)a{b}^k+\left(\begin{array}{l}k\\ k\end{array}\right)b^{k+1}\end{array}$

(e)

To determine

To calculate: The addition of binomial sums in brackets as per the results of Exercise 84.

It is provided that

$\begin{array}{c}{\left(a+b\right)}^{k+1}=\left(\begin{array}{l}k\\ 0\end{array}\right)a^{k+1}+\left[\left(\begin{array}{l}k\\ 0\end{array}\right)+\left(\begin{array}{l}k\\ 1\end{array}\right)\right]a^{k}b+\left[\left(\begin{array}{l}k\\ 1\end{array}\right)+\left(\begin{array}{l}k\\ 2\end{array}\right)\right]a^{k-1}{b}^2+\left[\left(\begin{array}{l}k\\ 2\end{array}\right)+\left(\begin{array}{l}k\\ 3\end{array}\right)\right]a^{k-2}{b}^3\\ +\cdots +\left[\left(\begin{array}{c}k\\ k-1\end{array}\right)+\left(\begin{array}{l}k\\ k\end{array}\right)\right]a{b}^k+\left(\begin{array}{l}k\\ k\end{array}\right)b^{k+1}\end{array}$

(f)

To determine

The resultant statement that must be proved. The statement can be obtained by substituting the results of $\left(\begin{array}{l}k\\ 0\end{array}\right)=\left(\begin{array}{c}k+1\\ 0\end{array}\right)$(why?) and $\left(\begin{array}{l}k\\ k\end{array}\right)=\left(\begin{array}{c}k+1\\ k+1\end{array}\right)$ and the results of part e in to the equation of part d.