Chapter ISG, Problem 7.10.7P

a.

To determine

To explain: The given sequence is neither arithmetic nor geometric.

Answer to Problem 7.10.7P

The given sequence neither has common differences nor common ratios, so the given sequence is neither arithmetic nor geometric.

Explanation of Solution

Given Information: Given sequence is $a_1=1,a_n=3a_{n-1}+2$.

Calculation:

Putting $n=2,3,4,\mathrm{....}$ in $a_n=3a_{n-1}+2$ ,

  $a_2=3a_1+2=3(1)+2=5$

  $a_3=3a_2+2=3(5)+2=17$

  $a_4=3a_3+2=3(17)+2=53$ and so on.

So, the given sequence is $1,5,17,53,\mathrm{....}$

Now, $a_2-a_1=5-1=4$ and $a_3-a_2=17-5=12$.

Since the terms do not have common differences, so the given sequence is not arithmetic.

Also, $\frac{a_2}{a_1}=\frac{5}{1}=5$ , $\frac{a_3}{a_2}=\frac{17}{5}$

Since the terms do not have common ratios, so the given sequence is not geometric.

Thus, the given sequence is neither arithmetic nor geometric.

b.

To determine

To write: Formulas for $a_n$ in terms of $a_{n-2}$ and of $a_{n-3}$.

Answer to Problem 7.10.7P

The formulas for $a_n$ in terms of $a_{n-2}$ and of $a_{n-3}$ are $a_n=9a_{n-2}+8\&a_n=27a_{n-3}+26$ respectively.

Explanation of Solution

Given Information: Given sequence is $a_1=1,a_n=3a_{n-1}+2$.

Calculation:

  $a_n=3a_{n-1}+2$ -(i)

Replacing $n$ by $n-1$ ,

  $a_{n-1}=3a_{n-2}+2$

Putting this value of $a_{n-1}$ in (i),

  $a_n=3(3a_{n-2}+2)+2$

  $\Rightarrow a_n=9a_{n-2}+8$ -(ii)

Replacing $n$ by $n-2$ in (i),

  $a_{n-2}=3a_{n-3}+2$

Putting this value of $a_{n-2}$ in (ii),

  $a_n=9(3a_{n-3}+2)+8$

  $\Rightarrow a_n=27a_{n-3}+26$

Thus, the formulas for $a_n$ in terms of $a_{n-2}$ and of $a_{n-3}$ are $a_n=9a_{n-2}+8\&a_n=27a_{n-3}+26$ respectively.

c.

To determine

To write: An explicit formula for the given sequence in terms of a sum of powers of $3$.

Answer to Problem 7.10.7P

An explicit formula for the given sequence in terms of a sum of powers of $3$ is:

  $a_n=3^{n-1}+(3^{n-2}+3^{n-3}+\mathrm{....}+3+1)2$.

Explanation of Solution

Given Information: Given sequence is $a_1=1,a_n=3a_{n-1}+2$.

Calculation:

  $\begin{array}{l}{a}_1=1\\ a_2=3a_1+2=3+2\\ a_3=3a_2+2=3(3+2)+2=3^2+(3+1)2\\ a_4=3a_3+2=3(3^2+(3+1)2)+2=3^3+(3^2+3+1)2\end{array}$

Generalizing the pattern,

  $a_n=3^{n-1}+(3^{n-2}+3^{n-3}+\mathrm{....}+3+1)2$

Thus, an explicit formula for the given sequence in terms of a sum of powers of $3$ is:

  $a_n=3^{n-1}+(3^{n-2}+3^{n-3}+\mathrm{....}+3+1)2$.

d.

To determine

To write: A shorter explicit formula for the given sequence, using $k^n-1=(k-1)(k^{n-1}+k^{n-2}+\mathrm{...}+k+1)$.

Answer to Problem 7.10.7P

A shorter explicit formula is $a_n=2(3^{n-1})-1$

Explanation of Solution

Given Information: Given sequence is $a_1=1,a_n=3a_{n-1}+2$ and from part (c), an explicit formula for the given sequence is $a_n=3^{n-1}+(3^{n-2}+3^{n-3}+\mathrm{....}+3+1)2$.

Calculation:

Putting $k=3$ in $k^n-1=(k-1)(k^{n-1}+k^{n-2}+\mathrm{...}+k+1)$ and replacing $n$ by $n-1$ ,

  $3^{n-1}-1=(3-1)(3^{n-2}+3^{n-3}+\mathrm{...}+3+1)$

  $=2(3^{n-2}+3^{n-3}+\mathrm{...}+3+1)$

So, $a_n=3^{n-1}+3^{n-1}-1$

  $=2(3^{n-1})-1$

Thus, a shorter explicit formula is $a_n=2(3^{n-1})-1$.