Chapter ISG, Problem 7.9.4P

a.

To determine

To calculate which sequence has a greater common ratios.

Answer to Problem 7.9.4P

Sequence one has a greater common ratio.

Explanation of Solution

Given information: Sequence 1’s equation is,

  $a_n=512{(0.5)}^x$

Sequence 2’s terms,

  $b_3=7168$

And $b_7=28$

Concept Used:

Geometric sequence formula is,

  $a_n=a{r}^{n-1}$

Calculation:

For sequence 1, comparing to the equation. Common ratio is,

  $r=0.5$

For sequence 2,

Substituting the value of third term in the geometric sequence formula,

  $7168=a{r}^{3-1}$

On solving,

  $7168=a{r}^2$ _________________(1)

Substituting the value of 7^th term in the geometric sequence formula,

  $28=a{r}^{7-1}$

On solving,

  $28=a{r}^6$

Substituting (1),

  $28=7168r^4$

Cross-multiplying both sides,

  $r^4=\frac{28}{7168}$

So,

  $r=\frac{2.3003}{9.2013}$

On solving,

  $r=0.25$

Hence, common ratio for sequence 2 is 0.25

Comparing both sequence’s common ratio,

  $0.5>0.25$

Hence, Sequence one has a greater common ratio.

Conclusion:

Sequence one has a greater common ratio.

b.

To determine

To calculate initial term of both series.

Answer to Problem 7.9.4P

First term of sequence 1 is 512 and first term of sequence 2 is 114688

Explanation of Solution

Given information:

Sequence 1’s equation is,

  $a_n=512{(0.5)}^x$

Common ratio is 0.5

Sequence 2’s terms,

  $b_3=7168$

And $b_7=28$

Common ratio is 0.25

Formula Used:

Geometric sequence formula is,

  $a_n=a{r}^{n-1}$

Calculation:

For sequence 1, comparing to the equation. First term is,

  $a=512$

Hence, first term of sequence 1 is 512

For sequence 2,

Substituting the value of third term in the geometric sequence formula,

  $7168=a{(0.25)}^{3-1}$

So,

  $7168=0.0625a$

Cross-multiplying both sides,

  $\frac{7168}{0.0625}=a$

On solving,

  $a=114688$

Hence, first term of sequence 2 is 114688

Conclusion:

First term of sequence 1 is 512 and first term of sequence 2 is 114688