Chapter ISG, Problem 7.9.1P

a.

To determine

To find the function and 10^th term of the geometric sequence.

Answer to Problem 7.9.1P

The sequence is 0.5, 6, 72, 864, 10368.

The function is $f(x)=0.5{(12)}^{x-1}$ and the 10^th term is $2,579,890,176$

Explanation of Solution

Given information:

The given sequence is,

0.5, 6, c, d, e

Where, c and d are unknown.

Formula Used:

Common ratio is,

  $r=\frac{a2}{a1}$

Geometric sequence formula is,

  $a_n=a{(r)}^{n-1}$

Calculation:

For common ratio, substituting the values,

  $r=\frac{6}{0.5}$

On solving,

  $r=1.2$

For the function, substituting the values,

  $f(x)=0.5{(12)}^{x-1}$

For the 3^rd term, substituting the value of n,

  $f(3)=0.5{(12)}^{3-1}$

So,

  $f(3)=0.5{(12)}^2$

On solving,

  $f(3)=72$

For the 4^th term, substituting the value of n,

  $f(4)=0.5{(12)}^{4-1}$

So,

  $f(4)=0.5{(12)}^3$

On solving,

  $f(4)=864$

For the 5^th term, substituting the value of n,

  $f(5)=0.5{(12)}^{5-1}$

So,

  $f(5)=0.5{(12)}^4$

On solving,

  $f(5)=10,368$

Hence, the sequence is,

0.5, 6, 72, 864, 10368

For the 10^th term, substituting the value of n,

  $f(10)=0.5{(12)}^{10-1}$

So,

  $f(10)=0.5{(12)}^9$

On solving,

  $f(10)=2,579,890,176$

Conclusion:

The function is $f(x)=0.5{(12)}^{x-1}$ and the 10^th term is $2,579,890,176$ .

b.

To determine

To find the function and 10^th term of the geometric sequence.

Answer to Problem 7.9.1P

The sequence is 5, 10, 20, 40, 80

The function is $f(x)=5{(2)}^{x-1}$ and the 10^th term is $f(10)=2560$ .

Explanation of Solution

Given information:

The given sequence is,

a, 10, c, 40, e

Where a, c and e are unknown.

Formula Used:

Common ratio is,

  $r=\frac{a2}{a1}$

Geometric sequence formula is,

  $a_n=a{(r)}^{n-1}$

Calculation:

The second term is 10, hence, substituting the value,

  $10=a{(r)}^{2-1}$

Hence,

  $10=ar$ ___________________(1)

The fourth term is 40, hence, substituting the value,

  $40=a{(r)}^{4-1}$

Hence,

  $40=a{r}^3$

Substituting the value of (1),

  $40=(10)r^2$

Cross-multiplying,

  $\frac{40}{10}=r^2$

On solving,

  $r^2=4$

Square rooting both side,

  $\sqrt{r^2}=\sqrt{4}$

On solving,

  $r=2$

Hence, common difference is 2.

For the first term, substituting the values in (1),

  $10=a(2)$

Cross-multiplying,

  $a=\frac{10}{2}$

On solving,

  $a=5$

For the equation, substituting the values,

  $f(x)=5{(2)}^{x-1}$

For the 3^rd term, substituting the value of n,

  $f(3)=5{(2)}^{3-1}$

So,

  $f(3)=5{(2)}^2$

On solving,

  $f(3)=20$

For the 5^th term, substituting the value of n,

  $f(5)=5{(2)}^{5-1}$

So,

  $f(5)=5{(2)}^4$

On solving,

  $f(5)=80$

Hence, the sequence is,

5, 10, 20, 40, 80

For the 10^th term, substituting the value of n,

  $f(10)=5{(2)}^{10-1}$

So,

  $f(10)=5{(2)}^9$

On solving,

  $f(10)=2560$

Conclusion:

The function is $f(x)=5{(2)}^{x-1}$ and the 10^th term is $f(10)=2560$ .

c.

To determine

To find the function and 10^th term of the geometric sequence.

Answer to Problem 7.9.1P

The sequence is 6.159, -4, 2.598, -1.687, 1.6875

The function is $f(x)=6.159{(-0.6495)}^{x-1}$ and the 10^th term is $f(10)=0.1267$ .

Explanation of Solution

Given information:

The given sequence is,

a, -4, c, d, 1.6875

Where a, c and d are unknown.

Formula Used:

Common ratio is,

  $r=\frac{a2}{a1}$

Geometric sequence formula is,

  $a_n=a{(r)}^{n-1}$

Calculation:

The second term is -4, hence, substituting the value,

  $-4=a{(r)}^{2-1}$

Hence,

  $-4=ar$ ___________________(1)

The fifth term is 1.6875, hence, substituting the value,

  $1.6875=a{(r)}^{5-1}$

Hence,

  $1.6875=a{r}^4$

Substituting the value of (1),

  $1.6875=(-4)r^3$

Cross-multiplying,

  $\frac{1.6875}{-4}=r^3$

On solving,

  $r^3=-0.421875$

Cube rooting both side,

  $r=-0.6495$

Hence, common difference is -0.6495

For the first term, substituting the values in (1),

  $-4=a(-0.6495)$

Cross-multiplying,

  $a=\frac{-4}{-0.6495}$

On solving,

  $a=6.159$

For the equation, substituting the values,

  $f(x)=6.159{(-0.6495)}^{x-1}$

For the 3^rd term, substituting the value of n,

  $f(3)=6.159{(-0.6495)}^{3-1}$

So,

  $f(3)=6.159{(-0.6495)}^{2-1}$

On solving,

  $f(3)=2.598$

For the 4^th term, substituting the value of n,

  $f(4)=6.159{(-0.6495)}^{4-1}$

So,

  $f(4)=6.159{(-0.6495)}^3$

On solving,

  $f(4)=-1.687$

Hence, the sequence is,

6.159, -4, 2.598, -1.687, 1.6875

For the 10^th term, substituting the value of n,

  $f(10)=6.159{(-0.6495)}^{10-1}$

So,

  $f(10)=6.159{(-0.6495)}^9$

On solving,

  $f(10)=0.1267$

Conclusion:

The function is $f(x)=6.159{(-0.6495)}^{x-1}$ and the 10^th term is $f(10)=0.1267$ .