Sequences
Find the first five terms:
a_1 = \frac{1}{2}; a_n = a_{n-1} + \frac{3}{2}, n \geq 2
frac{1}{2}, 2, 3\frac{1}{2}, 5, 6\frac{1}{2}
Simplify:
(\frac{4}{7}x)^2 (49x)(17y)(\frac{1}{34}y^5)
8x^3y^6
Express the answer in both standard form and scientific notation.
(3 * 10^3)^2
9*10^6; 9,000,000
Determine if the sequence is arithmetic, geometric, or neither. Write a formula if possible.
2, -3, 4.5, -6.75, ...
a_n = 2 * (-1.5)^{n-1}
Simplify:
\frac{(x^2y^{-4}z^5)^0}{(2x^-5y^7z^5)^{-2}}
\frac{4y^{14}z^{10}}{x^{10}}
Express the answer in both standard form and scientific notation.
\frac{2 * 10^{-8}}{4 * 10^{-2}}
5 *10^{-7}; 0.0000005
Write an explicit equation and find the 10th term of the sequence: 6, -24, 96, ...
a_n = 6 *(-4)^{n-1}, -1,572,864
Simplify:
(\frac{-3x^{-6}y^{-1}z^{-2}}{6x^{-2}yz^{-5}})^{-2}
\frac{4x^8y^4}{z^6}
Paige invested $1200 at a rate of 5.75 % compounded quarterly. Determine the value of her investment after 7 years.
$1,789.54
Write a recursive formula for the sequence: 81, 27, 9, 3, ...
a_1 = 81; a_n = \frac{1}{3}a_{n-1}, n\geq 2
Simplify:
9^{-\frac{3}{2}}
\frac{1}{27}
Camilo purchased a rare coin from a dealer for $300. The value increases at 5% each year. Write an equation to model the value over time. Use the equation to find the value after 5 years.
y =300*(1.05)^x, \$ 382.88
Write a recursive formula for the explicit formula below.
a_n = 3(4)^{n-1}
a_1 = 3 ; a_n = 4* a_{n-1}, n \geq 2
Simplify and write in radical form:
(y^{\frac{1}{2}})^{\frac{2}{3}}
root(3){y}
Leonardo purchases a car for $18,995. It depreciates at a rate of 18% per year. Create an equation to represent the car's value over time. After 6 years how much is it worth?
y = 18995*(0.82)^x, \$5,774.61
Write an explicit formula for the recursive formula below.
a_1 = 38 ; a_n = \frac{1}{2}a_{n-1}, n \geq 2
a_n = 38 (\frac{1}{2})^{n-1}
Solve for x:
25^x = \frac{1}{125}
x = \frac{-3}{2}
Jin's investment of $4,500 has been losing value at a rate of 2.5 % each year. Create an equation to represent the value of the investment over time and find the value after 5 years.
y = 4500 (1- 0.025)^x, \$3,964.93
Danielle's parents have offered her two different options to earn her allowance for a 9 week period over the summer. She can either get paid $30 each week or $1 the first week, $2 the second week, $4 the third week, and so on. Find the total she would get paid with each option. Which should she choose?
$270 with the first and $511 with the second. She should choose the second one because she will earn $241 more.
Solve for x:
4^x = 32
x = \frac{5}{2}
Determine the growth rate (as a percent) of a population that quadruples every year.
300%