Question 1

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-A small business owner made $50,000 the first year he owned his store and made an additional 2% over the previous year in each subsequent year. Find how much he made during his fourth year of business. Find his total earnings during the first four years. (Round to the nearest cent, if necessary.)

Accepted Answer

The earnings each year form a geometric sequence with the first term $50,000 and a common ratio of 1.02 (reflecting a 2% increase each year). The fourth year's earnings can be found using the formula for the nth term of a geometric sequence, $a_n = a_1 \cdot r^{(n-1)}$, where $a_1 = 50,000$, $r = 1.02$, and $n = 4$. This calculation gives the earnings for the fourth year as approximately $53,060.40. The total earnings over the four years can be found using the formula for the sum of the first n terms of a geometric sequence, $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$, which gives approximately $206,080.40 for the first four years.

Question 2

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-As Sunee improves her algebra skills, she takes 0.9 times as long to complete each homework assignment as she took to complete the preceeding assignment. If it took her 60 minutes to complete her first assignment, find how long it took her to complete the fifth assignment. Find the total time she took to complete her first five homework assignments. (Round to the nearest minute.)

Accepted Answer

The sum of the first n terms of a geometric sequence is given by the formula:$$S_n = a_1(1 - r^n) / (1 - r)$$where a_1 is the first term, r is the common ratio, and n is the number of terms.In this case, a_1 = 60 and r = 0.9. We want to find the sum of the first 5 terms, so n = 5.Plugging these values into the formula, we get:$$S_5 = 60(1 - 0.9^5) / (1 - 0.9)$$$$S_5 = 60(1 - 0.59049) / (0.1)$$$$S_5 = 60(0.40951) / (0.1)$$$$S_5 = 245.706$$Rounding to the nearest minute, we get 246 minutes.The fifth term of the sequence is given by the formula:$$a_n = a_1 * r^(n-1)$$Plugging in the values we know, we get:$$a_5 = 60 * 0.9^(5-1)$$$$a_5 = 60 * 0.9^4$$$$a_5 = 60 * 0.6561$$$$a_5 = 39.366$$Rounding to the nearest minute, we get 39 minutes.

Question 3

Use the Formula for the Sum of an Infinite Geometric Series
-$$\frac { 1 } { 3 } - 1 + 3 - \ldots$$

Accepted Answer

The formula for the sum of an infinite geometric series is $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. For the series to converge, $|r| < 1$. In this series, the first term $a = \frac{1}{3}$ and the common ratio $r = -3$. Since $|r| = 3 > 1$, the series does not converge, and thus the sum does not exist.

Question 4

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-A job pays a salary of 30,000 the first year. During the next 7 years, the salary increases by 4% each year. What is the salary for the 8th year? What is the total salary over the 8-year period? (Round to the nearest cent.)

Accepted Answer

The salary for the 8th year is calculated using the formula for the nth term of a geometric sequence: $a_n = a_1 \cdot r^{(n-1)}$, where $a_1 = 30,000$, $r = 1.04$ (4% increase), and $n = 8$. This gives $a_8 = 30,000 \cdot 1.04^{7} \approx 39,477.95$. The total salary over the 8-year period is found using the sum formula for a geometric series: $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$, which gives $S_8 = 30,000 \cdot \frac{1 - 1.04^8}{1 - 1.04} \approx 276,426.79$.

Question 5

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-$$\sum _ { \mathrm { i } = 1 } ^ { 5 } \frac { 4 } { 3 } \cdot 2 ^ { \mathrm { i } }$$

Accepted Answer

The sum of the first n terms of a geometric sequence is given by $S_n = \frac{a(1 - r^n)}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. Here, $a = \frac{4}{3} \cdot 2^1 = \frac{8}{3}$, $r = 2$, and $n = 5$. Plugging these values into the formula gives $S_5 = \frac{\frac{8}{3}(1 - 2^5)}{1 - 2} = \frac{\frac{8}{3}(1 - 32)}{-1} = \frac{8}{3} \cdot 31 = \frac{248}{3}$.

Question 6

Use the Formula for the Sum of the First n Terms of a Geometric Sequence
-Find the sum of the first 11 terms of the geometric sequence: $\frac { 1 } { 2 } , 1,2,4,8 , \ldots$

Accepted Answer

The sum of the first n terms of a geometric sequence is given by $S_n = \frac{a(1 - r^n)}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. Here, $a = \frac{1}{2}$, $r = 2$, and $n = 11$. Plugging these values into the formula gives $S_{11} = \frac{\frac{1}{2}(1 - 2^{11})}{1 - 2} = \frac{\frac{1}{2}(1 - 2048)}{-1} = \frac{2047}{2}$.

Question 7

Find the Value of an Annuity
-Looking ahead to retirement, you sign up for automatic savings in a fixed-income 401K plan that pays 5.5% per year compounded annually. You plan to invest $2000 at the end of each year for the next 25 years. How much will your account have in it at the end of 25 years?

Accepted Answer

The future value of an annuity is calculated using the formula:FV = P * [(1 + r)^n - 1] / rwhere:P is the periodic paymentr is the annual interest raten is the number of periodsIn this case, P = $2000, r = 5.5%, and n = 25. Plugging these values into the formula, we get:FV = 2000 * [(1 + 0.055)^25 - 1] / 0.055= 2000 * [2.853 - 1] / 0.055= 2000 * 1.853 / 0.055= $102,305Therefore, the correct answer is A) $102,305.

Question 8

Use the Formula for the Sum of an Infinite Geometric Series
-$$\sum _ { i = 1 } ^ { \infty } 10 ( - 0.2 ) ^ { \mathrm { i } - 1 }$$

Accepted Answer

The formula for the sum of an infinite geometric series is $\frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Here, $a = 10$ and $r = -0.2$. Plugging these values into the formula gives $\frac{10}{1 - (-0.2)} = \frac{10}{1.2} = \frac{25}{3}$.

Question 9

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-$$\sum _ { i = 1 } ^ { 3 } \left( \frac { 2 } { 3 } \right) ^ { i + 1 }$$

Accepted Answer

The sum of the first n terms of a geometric sequence is given by $S_n = \frac{a(1 - r^n)}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. For this sequence, $a = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$, $r = \frac{2}{3}$, and $n = 3$. Plugging these values into the formula gives $S_3 = \frac{\frac{4}{9}(1 - (\frac{2}{3})^3)}{1 - \frac{2}{3}} = \frac{\frac{4}{9}(1 - \frac{8}{27})}{\frac{1}{3}} = \frac{\frac{4}{9} \cdot \frac{19}{27}}{\frac{1}{3}} = \frac{76}{81}$.

Question 10

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-$$\sum _ { i = 1 } ^ { 5 } 3 \cdot 4 i$$

Accepted Answer

The given series is a geometric series with the first term $ a = 12 $ and common ratio $ r = 4 $. The sum of the first $ n $ terms of a geometric series is given by $ S_n = a \frac{r^n - 1}{r - 1} $. For $ n = 5 $, the sum is $ S_5 = 12 \frac{4^5 - 1}{4 - 1} = 12 \times \frac{1024 - 1}{3} = 12 \times 341 = 4092 $.

Question 11

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-$$\sum _ { \mathrm { i } = 1 } ^ { 5 } 4 ( - 2 ) ^ { \mathrm { i } }$$

Accepted Answer

The sum of the first n terms of a geometric sequence is given by the formula $$S_n = \frac{a_1(1 - r^n)}{1 - r}$$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. Here, $a_1 = 4(-2)^1 = -8$, $r = -2$, and $n = 5$. Plugging these values into the formula gives $$S_5 = \frac{-8(1 - (-2)^5)}{1 - (-2)} = \frac{-8(1 + 32)}{3} = \frac{-8 \cdot 33}{3} = -8 \cdot 11 = -88$$.

Question 12

Use the Formula for the Sum of an Infinite Geometric Series
-$3 - 1 + \frac { 1 } { 3 } - \frac { 1 } { 9 } + \ldots$

Accepted Answer

The given series is an infinite geometric series with the first term $a = 3$ and common ratio $r = -\frac{1}{3}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$ when $|r| < 1$. Substituting $a = 3$ and $r = -\frac{1}{3}$, we get $S = \frac{3}{1 - (-\frac{1}{3})} = \frac{3}{\frac{4}{3}} = \frac{9}{4}$.

Question 13

Find the Value of an Annuity
-Kurt deposits $150 each month into an account paying annual interest of 6% compounded monthly. How much will his account have in it at the end of 9 years?

Accepted Answer

The correct answer is A.The future value of an annuity is given by the formula:FV = P * ((1 + r/n)^(n*t) - 1) / (r/n)where:P is the monthly depositr is the annual interest raten is the number of times per year that the interest is compoundedt is the number of yearsIn this case, we have:P = $150r = 6% = 0.06n = 12 (since the interest is compounded monthly)t = 9Plugging these values into the formula, we get:FV = $150 * ((1 + 0.06/12)^(12*9) - 1) / (0.06/12)= $150 * ((1.005)^108 - 1) / 0.005= $150 * (2.1411 - 1) / 0.005= $150 * 214.11 / 0.005= $21,411Therefore, Kurt's account will have $21,411 in it at the end of 9 years.

Question 14

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
-$$\sum _ { i = 1 } ^ { 8 } \left( \frac { 4 } { 3 } \right) ^ { i }$$

Accepted Answer

The sum of the first n terms of a geometric sequence is given by the formula $$S_n = \frac{a(1 - r^n)}{1 - r}$$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. Here, $a = \left(\frac{4}{3}\right)^1 = \frac{4}{3}$, $r = \frac{4}{3}$, and $n = 8$. Plugging these values into the formula gives $$S_8 = \frac{\frac{4}{3}(1 - (\frac{4}{3})^8)}{1 - \frac{4}{3}} = \frac{\frac{4}{3}(1 - \frac{65536}{6561})}{-\frac{1}{3}} = \frac{4(6561 - 65536)}{-2187} = \frac{235,900}{6561}$$.

Question 15

Use the Formula for the Sum of an Infinite Geometric Series
-$75 + 15 + 3 + \frac { 3 } { 5 } + \ldots$

Accepted Answer

The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Here, $a = 75$ and $r = \frac{15}{75} = \frac{1}{5}$. Thus, $S = \frac{75}{1 - \frac{1}{5}} = \frac{75}{\frac{4}{5}} = \frac{75 \times 5}{4} = \frac{375}{4}$.

Question 16

Use the Formula for the Sum of an Infinite Geometric Series
-$$5 + \frac { 5 } { 2 } + \frac { 5 } { 4 } + \frac { 5 } { 8 } + \ldots$$

Accepted Answer

The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Here, $a = 5$ and $r = \frac{1}{2}$. Thus, $S = \frac{5}{1 - \frac{1}{2}} = \frac{5}{\frac{1}{2}} = 10$.

Question 17

Find the Value of an Annuity
-To save for retirement, you decide to deposit $2250 into an IRA at the end of each year for the next 40 years. If the interest rate is 8% per year compounded annually, find the value of the IRA after 40 years.

Accepted Answer

The future value of an annuity is given by the formula:FV = P * ((1 + r)^n - 1) / rwhere:P is the annual paymentr is the annual interest raten is the number of yearsIn this case, we have:P = $2250r = 8% = 0.08n = 40Plugging these values into the formula, we get:FV = $2250 * ((1 + 0.08)^40 - 1) / 0.08= $2250 * (14.207 - 1) / 0.08= $2250 * 13.207 / 0.08= $582,877Therefore, the value of the IRA after 40 years is $582,877.

Question 18

Find the Value of an Annuity
-Domenica invests $200 each quarter in a fixed-interest mutual fund paying annual interest of 6% compounded quarterly. How much will her account have in it at the end of 14 years?

Accepted Answer

The answer of Find the Value of an Annuity
-Domenica invests...

Question 19

Use the Formula for the Sum of an Infinite Geometric Series
-$- 10 - 5 - \frac { 5 } { 2 } - \frac { 5 } { 4 } - \ldots$

Accepted Answer

The series is an infinite geometric series with the first term $a = -10$ and common ratio $r = \frac{1}{2}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, so substituting $a = -10$ and $r = \frac{1}{2}$ gives $S = \frac{-10}{1 - \frac{1}{2}} = \frac{-10}{\frac{1}{2}} = -20$.

Question 20

Use the Formula for the Sum of the First n Terms of a Geometric Sequence
-Find the sum of the first 14 terms of the geometric sequence: $\frac { 1 } { 4 } , - \frac { 3 } { 4 } , \frac { 9 } { 4 } , - \frac { 27 } { 4 } , \frac { 81 } { 4 } , \ldots$

Accepted Answer

The first term is $\frac { 1 } { 4 } $ and the common ratio is -3. So, the sum of the first 14 terms is:$$S_{14} = \frac { \frac { 1 } { 4 } (1 - (-3)^{14}) } { 1 - (-3) } = \frac { \frac { 1 } { 4 } (1 - 4782969) } { 4 } = - \frac { 597871 } { 2 }$$

Quiz 11: Sequences, Induction, and Probability

Use the Formula for the Sum of an Infinite Geometric Series - $\frac { 1 } { 3 } - 1 + 3 - \ldots$

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. - $\sum _ { \mathrm { i } = 1 } ^ { 5 } \frac { 4 } { 3 } \cdot 2 ^ { \mathrm { i } }$

Use the Formula for the Sum of the First n Terms of a Geometric Sequence -Find the sum of the first 11 terms of the geometric sequence: $\frac { 1 } { 2 } , 1,2,4,8 , \ldots$

Find the Value of an Annuity -Looking ahead to retirement, you sign up for automatic savings in a fixed-income 401K plan that pays 5.5% per year compounded annually. You plan to invest $2000 at the end of each year for the next 25 years. How much will your account have in it at the end of 25 years?

Use the Formula for the Sum of an Infinite Geometric Series - $\sum _ { i = 1 } ^ { \infty } 10 ( - 0.2 ) ^ { \mathrm { i } - 1 }$

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. - $\sum _ { i = 1 } ^ { 3 } \left( \frac { 2 } { 3 } \right) ^ { i + 1 }$

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. - $\sum _ { i = 1 } ^ { 5 } 3 \cdot 4 i$

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. - $\sum _ { \mathrm { i } = 1 } ^ { 5 } 4 ( - 2 ) ^ { \mathrm { i } }$

Use the Formula for the Sum of an Infinite Geometric Series - $3 - 1 + \frac { 1 } { 3 } - \frac { 1 } { 9 } + \ldots$

Find the Value of an Annuity -Kurt deposits $150 each month into an account paying annual interest of 6% compounded monthly. How much will his account have in it at the end of 9 years?

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. - $\sum _ { i = 1 } ^ { 8 } \left( \frac { 4 } { 3 } \right) ^ { i }$

Use the Formula for the Sum of an Infinite Geometric Series - $75 + 15 + 3 + \frac { 3 } { 5 } + \ldots$

Use the Formula for the Sum of an Infinite Geometric Series - $5 + \frac { 5 } { 2 } + \frac { 5 } { 4 } + \frac { 5 } { 8 } + \ldots$

Find the Value of an Annuity -To save for retirement, you decide to deposit $2250 into an IRA at the end of each year for the next 40 years. If the interest rate is 8% per year compounded annually, find the value of the IRA after 40 years.

Find the Value of an Annuity -Domenica invests $200 each quarter in a fixed-interest mutual fund paying annual interest of 6% compounded quarterly. How much will her account have in it at the end of 14 years?

Use the Formula for the Sum of an Infinite Geometric Series - $- 10 - 5 - \frac { 5 } { 2 } - \frac { 5 } { 4 } - \ldots$

Use the Formula for the Sum of the First n Terms of a Geometric Sequence -Find the sum of the first 14 terms of the geometric sequence: $\frac { 1 } { 4 } , - \frac { 3 } { 4 } , \frac { 9 } { 4 } , - \frac { 27 } { 4 } , \frac { 81 } { 4 } , \ldots$

Quiz 11: Sequences, Induction, and Probability

Use the Formula for the Sum of an Infinite Geometric Series - 13−1+3−…\frac { 1 } { 3 } - 1 + 3 - \ldots31​−1+3−…

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. - ∑i=1543⋅2i\sum _ { \mathrm { i } = 1 } ^ { 5 } \frac { 4 } { 3 } \cdot 2 ^ { \mathrm { i } }∑i=15​34​⋅2i

Use the Formula for the Sum of the First n Terms of a Geometric Sequence -Find the sum of the first 11 terms of the geometric sequence: 12,1,2,4,8,…\frac { 1 } { 2 } , 1,2,4,8 , \ldots21​,1,2,4,8,…

Find the Value of an Annuity -Looking ahead to retirement, you sign up for automatic savings in a fixed-income 401K plan that pays 5.5% per year compounded annually. You plan to invest $2000 at the end of each year for the next 25 years. How much will your account have in it at the end of 25 years?

Use the Formula for the Sum of an Infinite Geometric Series - ∑i=1∞10(−0.2) i−1\sum _ { i = 1 } ^ { \infty } 10 ( - 0.2 ) ^ { \mathrm { i } - 1 }∑i=1∞​10(−0.2) i−1

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. - ∑i=13(23) i+1\sum _ { i = 1 } ^ { 3 } \left( \frac { 2 } { 3 } \right) ^ { i + 1 }∑i=13​(32​) i+1

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. - ∑i=153⋅4i\sum _ { i = 1 } ^ { 5 } 3 \cdot 4 i∑i=15​3⋅4i

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. - ∑i=154(−2) i\sum _ { \mathrm { i } = 1 } ^ { 5 } 4 ( - 2 ) ^ { \mathrm { i } }∑i=15​4(−2) i

Use the Formula for the Sum of an Infinite Geometric Series - 3−1+13−19+…3 - 1 + \frac { 1 } { 3 } - \frac { 1 } { 9 } + \ldots3−1+31​−91​+…