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What are the first five terms of the geometric sequence ${a}_{1}=3,\text{}{a}_{n}=4\cdot {a}_{n-1}?$
$3,\text{}12,\text{}48,\text{}192,\text{}768$
Write a recursive formula for the geometric sequence $1,\text{}\frac{1}{3},\text{}\frac{1}{9},\text{}\frac{1}{27},\dots $
Write an explicit formula for the geometric sequence $-\frac{1}{5},\text{}-\frac{1}{15},\text{}-\frac{1}{45},\text{}-\frac{1}{135},\dots $
${a}_{n}=-\frac{1}{5}\cdot {\left(\frac{1}{3}\right)}^{n-1}$
How many terms are in the finite geometric sequence $-5,-\frac{5}{3},-\frac{5}{9},\dots ,-\frac{5}{59\text{,}049}?$
Use summation notation to write the sum of terms $\frac{1}{2}m+5$ from $m=0$ to $m=5.$
$\sum _{m=0}^{5}\left(\frac{1}{2}m+5\right).$
Use summation notation to write the sum that results from adding the number $13$ twenty times.
Use the formula for the sum of the first $n$ terms of an arithmetic series to find the sum of the first eleven terms of the arithmetic series 2.5, 4, 5.5, … .
${S}_{11}=110$
A ladder has $15$ tapered rungs, the lengths of which increase by a common difference. The first rung is 5 inches long, and the last rung is 20 inches long. What is the sum of the lengths of the rungs?
Use the formula for the sum of the first n terms of a geometric series to find ${S}_{9}$ for the series $12,\text{}6,\text{}3,\text{}\frac{3}{2},\dots $
${S}_{9}\approx 23.95$
The fees for the first three years of a hunting club membership are given in [link] . If fees continue to rise at the same rate, how much will the total cost be for the first ten years of membership?
Year | Membership Fees |
---|---|
1 | $1500 |
2 | $1950 |
3 | $2535 |
Find the sum of the infinite geometric series $\sum}_{k=1}^{\infty}45\cdot {(-\frac{1}{3})}^{k-1}.$
$S=\frac{135}{4}$
A ball has a bounce-back ratio of $\frac{3}{5}$ the height of the previous bounce. Write a series representing the total distance traveled by the ball, assuming it was initially dropped from a height of 5 feet. What is the total distance? ( Hint : the total distance the ball travels on each bounce is the sum of the heights of the rise and the fall.)
Alejandro deposits $80 of his monthly earnings into an annuity that earns 6.25% annual interest, compounded monthly. How much money will he have saved after 5 years?
$5,617.61
The twins Sarah and Scott both opened retirement accounts on their 21 ^{st} birthday. Sarah deposits $4,800.00 each year, earning 5.5% annual interest, compounded monthly. Scott deposits $3,600.00 each year, earning 8.5% annual interest, compounded monthly. Which twin will earn the most interest by the time they are $55$ years old? How much more?
How many ways are there to choose a number from the set $\text{\hspace{0.17em}}\{-10\text{,}-6\text{,}4\text{,}10\text{,}12\text{,}18\text{,}24\text{,}32\}\text{\hspace{0.17em}}$ that is divisible by either $4$ or $6?$
6
In a group of $20$ musicians, $12$ play piano, $7$ play trumpet, and $2$ play both piano and trumpet. How many musicians play either piano or trumpet?
How many ways are there to construct a 4-digit code if numbers can be repeated?
${10}^{4}=10\text{,}000$
A palette of water color paints has 3 shades of green, 3 shades of blue, 2 shades of red, 2 shades of yellow, and 1 shade of black. How many ways are there to choose one shade of each color?
In a group of $5$ freshman, $10$ sophomores, $3$ juniors, and $2$ seniors, how many ways can a president, vice president, and treasurer be elected?
A coffee shop has 7 Guatemalan roasts, 4 Cuban roasts, and 10 Costa Rican roasts. How many ways can the shop choose 2 Guatemalan, 2 Cuban, and 3 Costa Rican roasts for a coffee tasting event?
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