Final Exam Review

Sequences and Series

100

Is the sequence 2, 4, 8, 10, ... arithmetic, geometric, or neither?

Neither

200

Give a formula for the n^th term of the sequence

2, 5, 8, 11, 14, ...

a_n = 2 + 3(n - 1)

300

Give a formula for the n^th term of the sequence

4, 12, 36, 108, 324, ...

a_n = 4 * 3^n-1

400

Give a formula for the n^th term of the sequence

4/2, 9/4, 16/6, 25/8, 36/10, ...

a_n = (n+1)²/2n

500

Compute 1 + 2 + 3 + ... + 99 + 100

5050

Systems

100

Which of the following is a solution to the system of equations?

2x - 3y = 5

x + 2y = -1

(a) (1, 1)

(b) (1, -1)

(c) (3, 2)

(b)

200

Which of the following is a solution to the system of inequalities?

2x - 3y < 5

x + 2y > -1

(a) (1, 1)

(b) (-5, -2)

(c) (3, 2)

(a) and (c)

300

Solve the system

3x + 4y = -11

-x + y = 6

(-5, 1)

400

Solve the system

5x - 2y = 4

3x + 3y = 15

(2, 3)

500

Solve the system

2x² + y² = 17

x² + 2y² = 22

(2, 3), (-2, 3), (-2, -3), (2, -3)

Polynomials and Rational Expressions

100

Find the roots (with multiplicity) of

x⁴ - 3x³ - 10x²

0 (multiplicity 2)

-2 (multiplicity 1)

5 (multiplicity 1)

200

Fully simplify the expression

1/(x - 1) + x/(2x + 3)

(x² + x + 3)/((x - 1)(2x + 3))

300

Solve the equation

(w + 3)/(4w) + 1 = (w - 5)/w

w = -23

400

Divide x⁵ + 32 by x + 2

x⁴ - 2x³ +4x² - 8x + 16

500

Solve the inequality (x + 3)(2x - 4)(x - 5) > 0

(-3, 2) U (5, ∞)

Exponents and Logarithms

100

Compute log₃(1/27)

-3

200

Fully simplify the expression

log(2x - 3) + 4log(x + 1) - (1/2)log(x)

log((2x - 3)(x+1)⁴/(x^1/2))

300

Solve the equation

log₂(x - 31) = 5 - log₂(x)

x = 32

400

Find the inverse function f^-1(x) of the function

f(x) = log₃(2x - 1)

f^-1(x) = (1/2)*3^x + 1/2

500

Solve the equation

3^{6x + 5} = 5^2x

x = (5*ln(3))/(2*ln(5) - 6*ln(3))

Word Problems

100

A $2500 bond grows to $3729.56 in 10 years under continuous compounding. Find the interest rate. (round to the nearest whole percent)

4%

200

A long jumper leaves the ground at an angle of 20° at a speed of 11 m/s. The height of the jumper can be modeled by h(x) = -0.046x² + 0.346x, where h(x) is the jumper's height in meters, and x is the horizontal distance from the start point.

What is the jumper's maximum height? (round to 2 decimal places)

0.72 m

300

A pharmacist wants to mix a 30% saline solution with a 10% saline solution to get 200 mL of a 12% saline solution. How much of each solution should be used?

20 mL of the 30% solution and 180 mL of the 10% solution

400

The perimeter of a rectangle is 30 ft, and the area is 50 ft². What are the dimensions of the rectangle?

5 ft by 10 ft

500

The sum of the squares of two positive numbers is 145 and the difference of the squares of the numbers is 17. Find the numbers

8 and 9