Chapter ISG, Problem 7.10.2P

a.

To determine

To find recursive formula.

Answer to Problem 7.10.2P

The recursive formula is $a_n=a_{n-1}-7$

Explanation of Solution

Given information:

The explicit formula is,

  $a_n=12-7n$

Formula Used:

Arithmetic sequence is,

  $a_n=a+d(n-1)$

Common ratio is,

  $d=a2-a1$

Calculation:

Comparing the given formula to the arithmetic formula. The common difference is,

  $d=-7$

For recursive formula, substituting the values,

  $a_n=a_{n-1}+(-7)$

Can rewrite it as,

  $a_n=a_{n-1}-7$

Hence, the recursive formula is $a_n=a_{n-1}-7$ .

Conclusion:

The explicit formula is $a_n=3{(2)}^{n-1}$ and the recursive formula is

  $a_n=2a_{n-1}$ .

b.

To determine

To find recursive and explicit formula.

Answer to Problem 7.10.2P

The recursive formula is $a_n=a_{n-1}(-3)$ .

Explanation of Solution

Given information:

The sequence formula is,

  $-a_n=-2{(-3)}^n$

Formula Used:

Geometric sequence is,

  $a_n=a{(r)}^{n-1}$

Common ratio is,

  $r=\frac{a2}{a1}$

Calculation:

Comparing the given equation and geometric function formula,

The common ratio is,

  $r=-3$

For recursive formula, substituting the values

  $a_n=a_{n-1}(-3)$

Hence, the recursive formula is $a_n=a_{n-1}(-3)$ .

Conclusion:

The recursive formula is $a_n=a_{n-1}(-3)$ .