수열 전체 문제

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?

Rachael deposits $1,500 into a retirement fund each year. The fund earns 8.2% annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55? How much of that amount will be interest earned?

Write the first four terms of the sequence defined by the recursive formula ${a}_{1}=2,\,{a}_{n}={a}_{n-1}+n.$

Evaluate $\frac{6!}{(5-3)!3!}.$

Write the first four terms of the sequence defined by the explicit formula ${a}_{n}={10}^{n}+3.$

Write the first four terms of the sequence defined by the explicit formula ${a}_{n}=\frac{n!}{n(n+1)}.$

Is the sequence $\frac{4}{7},\frac{47}{21},\frac{82}{21},\frac{39}{7},\,...$ arithmetic? If so, find the common difference.

Is the sequence $2,4,8,16,\,...$ arithmetic? If so, find the common difference.