수열 전체 문제

For the following exercises, use the formula for the sum of the first $n$ terms of a geometric series to find the partial sum.

$\sum_{k=1}^{9}{2}^{k-1}$

For the following exercises, use the formula for the sum of the first $n$ terms of a geometric series to find the partial sum.

$\sum_{n=1}^{10}-2\cdot {\left(\frac{1}{2}\right)}^{n-1}$

For the following exercises, find the sum of the infinite geometric series.

$4+2+1+\frac{1}{2}...$

For the following exercises, find the sum of the infinite geometric series.

$-1-\frac{1}{4}-\frac{1}{16}-\frac{1}{64}...$

For the following exercises, find the sum of the infinite geometric series.

$\sum_{k=1}^{\infty }3\cdot {\left(\frac{1}{4}\right)}^{k-1}$

For the following exercises, find the sum of the infinite geometric series.

$\sum_{n=1}^{\infty }4.6\cdot {0.5}^{n-1}$

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

Deposit amount: $\text{\textdollar}50;$ total deposits: $60;$ interest rate: $5\%,$ compounded monthly

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

Deposit amount: $\text{\textdollar}150;$ total deposits: $24;$ interest rate: $3\%,$ compounded monthly

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

Deposit amount: $\text{\textdollar}450;$ total deposits: $60;$ interest rate: $4.5\%,$ compounded quarterly

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

Deposit amount: $\text{\textdollar}100;$ total deposits: $120;$ interest rate: $10\%,$ compounded semi-annually

The sum of terms $50-{k}^{2}$ from $k=x$ through $7$ is $115.$ What is x?

Write an explicit formula for ${a}_{k}$ such that $\sum_{k=0}^{6}{a}_{k}=189.$ Assume this is an arithmetic series.

Find the smallest value of n such that $\sum_{k=1}^{n}(3k-5)>100.$

How many terms must be added before the series $-1-3-5-7....$ has a sum less than $-75?$

Write $0.\overline{65}$ as an infinite geometric series using summation notation. Then use the formula for finding the sum of an infinite geometric series to convert $0.\overline{65}$ to a fraction.

To get the best loan rates available, the Coleman family want to save enough money to place 20% down on a $160,000 home. They plan to make monthly deposits of $125 in an investment account that offers 8.5% annual interest compounded semi-annually. Will the Colemans have enough for a 20% down payment after five years of saving? How much money will they have saved?

Karl has two years to save $\text{\textdollar}10,000$ to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a 4.2% annual interest rate that compounds monthly?

Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study for $1$ hour, and each successive day she will increase her study time by $30$ minutes. How many hours will Keisha have studied after one week?

A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?

A scientist places 50 cells in a petri dish. Every hour, the population increases by 1.5%. What will the cell count be after 1 day?