Question 1

Differentiate the series for $- \frac { 1 } { x }$ to find the power series for the function $f ( x ) = \frac { 6 } { x ^ { 2 } }$ .

Accepted Answer

A)  $6 \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ( x - 1 ) ^ { n - 1 }$ 
B)  $\sum _ { n = 1 } ^ { \infty } ( - 6 ) ^ { n } n ( x - 1 ) ^ { n - 1 }$ 
C)  $6 \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ( x - 1 ) ^ { n - 1 }$ 
D)  $\sum _ { n = 0 } ^ { \infty } ( - 6 ) ^ { n - 1 } n ( x - 1 ) ^ { n }$ 
E)  $\sum _ { n = 0 } ^ { \infty } ( - 6 ) ^ { n - 1 } n ( x - 1 ) ^ { n + 1 }$ 
A)  $6 \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ( x - 1 ) ^ { n - 1 }$ 
B)  $\sum _ { n = 1 } ^ { \infty } ( - 6 ) ^ { n } n ( x - 1 ) ^ { n - 1 }$ 
C)  $6 \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ( x - 1 ) ^ { n - 1 }$ 
D)  $\sum _ { n = 0 } ^ { \infty } ( - 6 ) ^ { n - 1 } n ( x - 1 ) ^ { n }$ 
E)  $\sum _ { n = 0 } ^ { \infty } ( - 6 ) ^ { n - 1 } n ( x - 1 ) ^ { n + 1 }$

Question 2

Write an expression for the apparent nth term of the sequence. (Assume that n begins with 1.) 9, 12, 15, 18, 21

Accepted Answer

A)  $a _ { n } = 6 n + 3$ 
B)  $a _ { n } = ( - 1 ) ^ { n } ( 3 n + 6 )$ 
C)  $a _ { n } = 3 n$ 
D)  $a _ { n } = 3 ^ { n } + 6$ 
E)  $a _ { n } = 3 n + 6$ 
A)  $a _ { n } = 6 n + 3$ 
B)  $a _ { n } = ( - 1 ) ^ { n } ( 3 n + 6 )$ 
C)  $a _ { n } = 3 n$ 
D)  $a _ { n } = 3 ^ { n } + 6$ 
E)  $a _ { n } = 3 n + 6$

Question 3

The value for which Newton's method fails for the function below is shown in the graph. Give the reason why the method fails. $y = - x ^ { 3 } + 3 x ^ { 2 } - x + 1$

Accepted Answer

A)  $\lim _ { x \rightarrow \infty } \left\{ \begin{array} { l } 
0 = x _ { 1 } = x _ { 3 } = \ldots \
- 1 = x _ { 2 } = x _ { 4 } = \ldots
\end{array} \right.$ 
B)  $f ^ { \prime } \left( x _ { 1 } \right) = 0$ 
C)  $f ^ { \prime } \left( x _ { 2 } \right) = 0$ 
D)  $f ^ { \prime \prime } \left( x _ { 2 } \right) = 0$ 
E)  $\lim _ { x \rightarrow \infty } \left\{ \begin{array} { l } 
1 = x _ { 1 } = x _ { 3 } = \ldots \
0 = x _ { 2 } = x _ { 4 } = \ldots
\end{array} \right.$ 
A)  $\lim _ { x \rightarrow \infty } \left\{ \begin{array} { l } 
0 = x _ { 1 } = x _ { 3 } = \ldots \
- 1 = x _ { 2 } = x _ { 4 } = \ldots
\end{array} \right.$ 
B)  $f ^ { \prime } \left( x _ { 1 } \right) = 0$ 
C)  $f ^ { \prime } \left( x _ { 2 } \right) = 0$ 
D)  $f ^ { \prime \prime } \left( x _ { 2 } \right) = 0$ 
E)  $\lim _ { x \rightarrow \infty } \left\{ \begin{array} { l } 
1 = x _ { 1 } = x _ { 3 } = \ldots \
0 = x _ { 2 } = x _ { 4 } = \ldots
\end{array} \right.$

Question 4

Bouncing Ball. A ball dropped from a height of 35 feet bounces to $1 / 2$ of its former height with each bounce. Find the total vertical distance that the ball travels.

Accepted Answer

A) 105 feet 
B) 140 feet 
C) 64 feet 
D) 53 feet 
E) 70 feet 
A) 105 feet 
B) 140 feet 
C) 64 feet 
D) 53 feet 
E) 70 feet

Question 5

Determine the convergence or divergence of the sequence $9 - \frac { 4 } { 5 ^ { n } }$ . If the sequence converges, use a symbolic algebra utility to find its limit.

Accepted Answer

A) 9 
B) 4 
C) 5 
D)  $- \infty$ 
E) The sequence diverges. 
A) 9 
B) 4 
C) 5 
D)  $- \infty$ 
E) The sequence diverges.

Question 6

You are in a boat 2 miles from the nearest point on the coast (see figure). You are to go to a point Q, which is 3 miles down the coast and 1 mile inland. You can row at 4 miles per hour and walk at 5 miles per hour. Toward what point on the coast should you row in order to reach point Q in the least time? Round your answer to three decimal places.

Accepted Answer

A) approximately 2.121 miles 
B) approximately 2.557 miles 
C) approximately 1.441 miles 
D) approximately 1.666 miles 
E) approximately 2.321 miles 
A) approximately 2.121 miles 
B) approximately 2.557 miles 
C) approximately 1.441 miles 
D) approximately 1.666 miles 
E) approximately 2.321 miles

Question 7

Determine whether the series $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }$ is a p-series.

Accepted Answer

A)  $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }$ is not a $p -$ series. 
B)  $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }$ is a $p -$ series. 
A)  $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }$ is not a $p -$ series. 
B)  $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }$ is a $p -$ series.

Question 8

Determine the convergence or divergence of the p-series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } }$ .

Accepted Answer

A) The series converges. 
B) The series diverges. 
A) The series converges. 
B) The series diverges.

Question 9

A factory is polluting a river such that at every mile down river from the factory an environmental expert finds 20% less pollutant than at the preceding mile. If the pollutant's concentration is 700 ppm (parts per million) at the factory, what is its concentration 15 miles down river?

Accepted Answer

A) 140.00 ppm 
B) 300.00 ppm 
C) 24.63 ppm 
D) 875.00 ppm 
E) 30.79 ppm 
A) 140.00 ppm 
B) 300.00 ppm 
C) 24.63 ppm 
D) 875.00 ppm 
E) 30.79 ppm

Question 10

Find the sum of the integers from 5 to 27.

Accepted Answer

A) 378 
B) 22 
C) 736 
D) 368 
E) 756 
A) 378 
B) 22 
C) 736 
D) 368 
E) 756

Question 11

Consider the sequence (An) whose nth term is given by An $= P \left[ 1 + \frac { r } { 12 } \right] ^ { n }$ where P is the principal, An is the amount of compound interest after n months, and r is the annual percentage rate. Write the first four terms of the sequence for P = $\$$ 8,000 and r = 0.04. Round your answer to two decimal places.

Accepted Answer

A) 8026.67, 8063.49, 8101.43, 8107.20 
B) 8027.67, 8063.49, 8101.43, 8135.19 
C) 8026.67, 8053.42, 8080.27, 8107.20 
D) 8027.67, 8063.49, 8080.27, 8107.20 
E) 8027.67, 8053.42, 8080.27, 8135.19 
A) 8026.67, 8063.49, 8101.43, 8107.20 
B) 8027.67, 8063.49, 8101.43, 8135.19 
C) 8026.67, 8053.42, 8080.27, 8107.20 
D) 8027.67, 8063.49, 8080.27, 8107.20 
E) 8027.67, 8053.42, 8080.27, 8135.19

Question 12

You accept a job that pays a salary of $\$$ 50,000 the first year. During the next 39 years, you will receive a 4% raise each year. What would be your total compensation over the 40-year period? Round your answer to the nearest integer.

Accepted Answer

A)  $\$$ 4,751,276 
B)  $\$$ 1,250,000 
C)  $\$$ 48,000 
D)  $\$$ 200,000 
E)  $\$$ 2,000 
A)  $\$$ 4,751,276 
B)  $\$$ 1,250,000 
C)  $\$$ 48,000 
D)  $\$$ 200,000 
E)  $\$$ 2,000

Question 13

Determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } ( 0.650 ) ^ { n }$ . Use a symbolic algebra utility to verify your result.

Accepted Answer

A) The series converges. 
B) The series diverges. 
A) The series converges. 
B) The series diverges.

Question 14

Determine whether the sequence is arithmetic. If so, find the common difference. 3, 9, 27, 81, 243

Accepted Answer

A) 3 
B) 3n 
C) 3n - 3n-1 
D) -3 
E) not arithmetic 
A) 3 
B) 3n 
C) 3n - 3n-1 
D) -3 
E) not arithmetic

Question 15

Apply Taylor's Theorem to find the power series centered at $c = 0$ for the function $f ( x ) = e ^ { 12 x }$ .

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n + 1 } } { ( n + 1 ) ! }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n } } { ( n + 1 ) ! }$ 
C)  $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n } } { n ! }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { ( 12 x ) ^ { n + 1 } } { n ! }$ 
E)  $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n + 1 } } { n ! }$ 
A)  $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n + 1 } } { ( n + 1 ) ! }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n } } { ( n + 1 ) ! }$ 
C)  $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n } } { n ! }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { ( 12 x ) ^ { n + 1 } } { n ! }$ 
E)  $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n + 1 } } { n ! }$

Question 16

Use a graphing utility to graph the first 10 terms of the sequence. $a _ { n } = - 2 ( 0.6 ) ^ { n - 1 }$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 17

Find the Taylor polynomials (centred at zero) of degree (a) 1, (b) 2, (c) 3, and (d) 4. $f ( x ) = \frac { x } { x + 1 }$

Accepted Answer

A)  $S _ { 1 } ( x ) = x , S _ { 2 } ( x ) = x - x ^ { 2 } , S _ { 3 } ( x ) = x - x ^ { 2 } + x ^ { 3 } , S _ { 4 } ( x ) = x - x ^ { 2 } + x ^ { 3 } - x ^ { 4 }$ 
B)  $S _ { 1 } ( x ) = x , S _ { 2 } ( x ) = x + x ^ { 2 } , S _ { 3 } ( x ) = x + x ^ { 2 } + x ^ { 3 } , S _ { 4 } ( x ) = x + x ^ { 2 } + x ^ { 3 } + x ^ { 4 }$ 
C)  $S _ { 1 } ( x ) = x , S _ { 2 } ( x ) = x - x ^ { 2 } , S _ { 3 } ( x ) = x - x ^ { 2 } - x ^ { 3 } , S _ { 4 } ( x ) = x - x ^ { 2 } - x ^ { 3 } - x ^ { 4 }$ 
D)  $S _ { 1 } ( x ) = x , S _ { 2 } ( x ) = x + x ^ { 2 } , S _ { 3 } ( x ) = x + x ^ { 2 } - x ^ { 3 } , S _ { 4 } ( x ) = x + x ^ { 2 } - x ^ { 3 } + x ^ { 4 }$ 
E)  $S _ { 1 } ( x ) = x , S _ { 2 } ( x ) = x + x ^ { 2 } , S _ { 3 } ( x ) = x + x ^ { 2 } - x ^ { 3 } , S _ { 4 } ( x ) = x + x ^ { 2 } + x ^ { 3 } - x ^ { 4 }$ 
A)  $S _ { 1 } ( x ) = x , S _ { 2 } ( x ) = x - x ^ { 2 } , S _ { 3 } ( x ) = x - x ^ { 2 } + x ^ { 3 } , S _ { 4 } ( x ) = x - x ^ { 2 } + x ^ { 3 } - x ^ { 4 }$ 
B)  $S _ { 1 } ( x ) = x , S _ { 2 } ( x ) = x + x ^ { 2 } , S _ { 3 } ( x ) = x + x ^ { 2 } + x ^ { 3 } , S _ { 4 } ( x ) = x + x ^ { 2 } + x ^ { 3 } + x ^ { 4 }$ 
C)  $S _ { 1 } ( x ) = x , S _ { 2 } ( x ) = x - x ^ { 2 } , S _ { 3 } ( x ) = x - x ^ { 2 } - x ^ { 3 } , S _ { 4 } ( x ) = x - x ^ { 2 } - x ^ { 3 } - x ^ { 4 }$ 
D)  $S _ { 1 } ( x ) = x , S _ { 2 } ( x ) = x + x ^ { 2 } , S _ { 3 } ( x ) = x + x ^ { 2 } - x ^ { 3 } , S _ { 4 } ( x ) = x + x ^ { 2 } - x ^ { 3 } + x ^ { 4 }$ 
E)  $S _ { 1 } ( x ) = x , S _ { 2 } ( x ) = x + x ^ { 2 } , S _ { 3 } ( x ) = x + x ^ { 2 } - x ^ { 3 } , S _ { 4 } ( x ) = x + x ^ { 2 } + x ^ { 3 } - x ^ { 4 }$

Question 18

A deposit of $2000 is made in an account that earns 6% interest compounded monthly. The balance in the account after n months is given by $A _ { n } = 2000 \left( 1 + \frac { 0.06 } { 12 } \right) ^ { n } , n = 1,2,3 , \mathrm {~K}$ Find the balance in the account after 11 years by finding the 132th term of the sequence. Round to the nearest penny.

Accepted Answer

A) $4,379,295.09 
B) $3863.23 
C) $3844.01 
D) $265,320.00 
E) $3882.54 
A) $4,379,295.09 
B) $3863.23 
C) $3844.01 
D) $265,320.00 
E) $3882.54

Question 19

The ordering and transportation cost C of the components used in manufacturing a product is given by $C = 200 \left( \frac { 300 } { x ^ { 2 } } + \frac { x } { x + 20 } \right)$ where C is measured in thousands of dollars and x is the order size in hundreds. Find the order size that minimizes the cost. Round your answer to the nearest unit.

Accepted Answer

A) 58 hundreds 
B) 56 hundreds 
C) 53 hundreds 
D) 55 hundreds 
E) 60 hundreds 
A) 58 hundreds 
B) 56 hundreds 
C) 53 hundreds 
D) 55 hundreds 
E) 60 hundreds

Question 20

Write an expression for the most apparent nth term of the sequence. (Assume that $n$ begins with 1.) $\frac { 1 } { 3 } , - \frac { 1 } { 9 } , \frac { 1 } { 27 } , - \frac { 1 } { 81 } , \frac { 1 } { 243 } , \mathrm {~K}$

Accepted Answer

A)  $a _ { n } = \frac { ( - 1 ) ^ { n } } { 3 ^ { n } }$ 
B)  $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { 3 ^ { n } }$ 
C)  $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { 2 ^ { n } }$ 
D)  $a _ { n } = \frac { ( - 1 ) ^ { n } } { 2 ^ { n } }$ 
E)  $a _ { n } = \frac { 1 } { 2 ^ { n } }$ 
A)  $a _ { n } = \frac { ( - 1 ) ^ { n } } { 3 ^ { n } }$ 
B)  $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { 3 ^ { n } }$ 
C)  $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { 2 ^ { n } }$ 
D)  $a _ { n } = \frac { ( - 1 ) ^ { n } } { 2 ^ { n } }$ 
E)  $a _ { n } = \frac { 1 } { 2 ^ { n } }$

Differentiate the series for $- \frac { 1 } { x }$ to find the power series for the function $f ( x ) = \frac { 6 } { x ^ { 2 } }$ .

Write an expression for the apparent nth term of the sequence. (Assume that n begins with 1.) 9, 12, 15, 18, 21

Bouncing Ball. A ball dropped from a height of 35 feet bounces to $1 / 2$ of its former height with each bounce. Find the total vertical distance that the ball travels.

Determine the convergence or divergence of the sequence $9 - \frac { 4 } { 5 ^ { n } }$ . If the sequence converges, use a symbolic algebra utility to find its limit.

Determine whether the series $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }$ is a p-series.

Determine the convergence or divergence of the p-series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } }$ .

A factory is polluting a river such that at every mile down river from the factory an environmental expert finds 20% less pollutant than at the preceding mile. If the pollutant's concentration is 700 ppm (parts per million) at the factory, what is its concentration 15 miles down river?

Find the sum of the integers from 5 to 27.

You accept a job that pays a salary of $\$$ 50,000 the first year. During the next 39 years, you will receive a 4% raise each year. What would be your total compensation over the 40-year period? Round your answer to the nearest integer.

Determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } ( 0.650 ) ^ { n }$ . Use a symbolic algebra utility to verify your result.

Determine whether the sequence is arithmetic. If so, find the common difference. 3, 9, 27, 81, 243

Apply Taylor's Theorem to find the power series centered at $c = 0$ for the function $f ( x ) = e ^ { 12 x }$ .

Use a graphing utility to graph the first 10 terms of the sequence. $a _ { n } = - 2 ( 0.6 ) ^ { n - 1 }$

Find the Taylor polynomials (centred at zero) of degree (a) 1, (b) 2, (c) 3, and (d) 4. $f ( x ) = \frac { x } { x + 1 }$

Write an expression for the most apparent nth term of the sequence. (Assume that $n$ begins with 1.) $\frac { 1 } { 3 } , - \frac { 1 } { 9 } , \frac { 1 } { 27 } , - \frac { 1 } { 81 } , \frac { 1 } { 243 } , \mathrm {~K}$

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Probability Web

Filters

Exam 16: Series and Taylor Polynomials Web

Differentiate the series for −1x- \frac { 1 } { x }−x1​ to find the power series for the function f(x)=6x2f ( x ) = \frac { 6 } { x ^ { 2 } }f(x)=x26​ .

Write an expression for the apparent nth term of the sequence. (Assume that n begins with 1.) 9, 12, 15, 18, 21

The value for which Newton's method fails for the function below is shown in the graph. Give the reason why the method fails. y=−x3+3x2−x+1y = - x ^ { 3 } + 3 x ^ { 2 } - x + 1y=−x3+3x2−x+1

Bouncing Ball. A ball dropped from a height of 35 feet bounces to 1/21 / 21/2 of its former height with each bounce. Find the total vertical distance that the ball travels.

Determine the convergence or divergence of the sequence 9−45n9 - \frac { 4 } { 5 ^ { n } }9−5n4​ . If the sequence converges, use a symbolic algebra utility to find its limit.

Determine whether the series ∑n=1∞n−4/5\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }∑n=1∞​n−4/5 is a p-series.

Determine the convergence or divergence of the p-series ∑n=1∞1n4\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } }∑n=1∞​n41​ .

A factory is polluting a river such that at every mile down river from the factory an environmental expert finds 20% less pollutant than at the preceding mile. If the pollutant's concentration is 700 ppm (parts per million) at the factory, what is its concentration 15 miles down river?

Find the sum of the integers from 5 to 27.

You accept a job that pays a salary of $\$$ 50,000 the first year. During the next 39 years, you will receive a 4% raise each year. What would be your total compensation over the 40-year period? Round your answer to the nearest integer.

Determine the convergence or divergence of the series ∑n=0∞(0.650)n\sum _ { n = 0 } ^ { \infty } ( 0.650 ) ^ { n }∑n=0∞​(0.650)n . Use a symbolic algebra utility to verify your result.

Determine whether the sequence is arithmetic. If so, find the common difference. 3, 9, 27, 81, 243

Apply Taylor's Theorem to find the power series centered at c=0c = 0c=0 for the function f(x)=e12xf ( x ) = e ^ { 12 x }f(x)=e12x .

Use a graphing utility to graph the first 10 terms of the sequence. an=−2(0.6)n−1a _ { n } = - 2 ( 0.6 ) ^ { n - 1 }an​=−2(0.6)n−1

Find the Taylor polynomials (centred at zero) of degree (a) 1, (b) 2, (c) 3, and (d) 4. f(x)=xx+1f ( x ) = \frac { x } { x + 1 }f(x)=x+1x​

Write an expression for the most apparent nth term of the sequence. (Assume that nnn begins with 1.) 13,−19,127,−181,1243, K\frac { 1 } { 3 } , - \frac { 1 } { 9 } , \frac { 1 } { 27 } , - \frac { 1 } { 81 } , \frac { 1 } { 243 } , \mathrm {~K}31​,−91​,271​,−811​,2431​, K

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Probability Web

Filters

Differentiate the series for $- \frac { 1 } { x }$ to find the power series for the function $f ( x ) = \frac { 6 } { x ^ { 2 } }$ .

Bouncing Ball. A ball dropped from a height of 35 feet bounces to $1 / 2$ of its former height with each bounce. Find the total vertical distance that the ball travels.

Determine the convergence or divergence of the sequence $9 - \frac { 4 } { 5 ^ { n } }$ . If the sequence converges, use a symbolic algebra utility to find its limit.

Determine whether the series $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }$ is a p-series.

Determine the convergence or divergence of the p-series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } }$ .

Determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } ( 0.650 ) ^ { n }$ . Use a symbolic algebra utility to verify your result.

Apply Taylor's Theorem to find the power series centered at $c = 0$ for the function $f ( x ) = e ^ { 12 x }$ .

Use a graphing utility to graph the first 10 terms of the sequence. $a _ { n } = - 2 ( 0.6 ) ^ { n - 1 }$

Find the Taylor polynomials (centred at zero) of degree (a) 1, (b) 2, (c) 3, and (d) 4. $f ( x ) = \frac { x } { x + 1 }$

Write an expression for the most apparent nth term of the sequence. (Assume that $n$ begins with 1.) $\frac { 1 } { 3 } , - \frac { 1 } { 9 } , \frac { 1 } { 27 } , - \frac { 1 } { 81 } , \frac { 1 } { 243 } , \mathrm {~K}$