13.4 Series and Their Notations

Use summation notation to write the sum.

Write the sum using summation notation: $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}$

Find the sum of the first n terms of an arithmetic sequence.

1. ⓐ Find the sum of the first 30 terms of the arithmetic sequence: 7, 10, 13, 13, 19,...
2. ⓑ Find the sum of the first 50 terms of the arithmetic sequence whose general term is $a_{n} = 2 n - 5$ .
3. ⓒ Find the sum
   $\sum_{i = 1}^{30} \left(6 i - 4\right)$

Evaluate $\sum_{k = 3}^{7} k^{2} .$

Evaluate $\sum_{k = 2}^{5} \left(3 k – 1\right) .$

Find the sum of each arithmetic series.

1. ⓐ $\text{5}\textrm{ }+\textrm{ }\text{8}\textrm{ }+\textrm{ }\text{11}\textrm{ }+\textrm{ }\text{14}\textrm{ }+\textrm{ }\text{17}\textrm{ }+\textrm{ }\text{20}\textrm{ }+\textrm{ }\text{23}\textrm{ }+\textrm{ }\text{26}\textrm{ }+\textrm{ }\text{29}\textrm{ }+\textrm{ }\text{32}$
2. ⓑ $\text{20}\textrm{ }+\textrm{ }\text{15}\textrm{ }+\textrm{ }\text{10}\textrm{ }+\ldots+\textrm{ }−\text{50}$
3. ⓒ $\sum_{k = 1}^{12} 3 k - 8$

$\text{1} .\text{4}\textrm{ }+\textrm{ }\text{1} .\text{6}\textrm{ }+\textrm{ }\text{1} .\text{8}\textrm{ }+\textrm{ }\text{2} .\text{0}\textrm{ }+\textrm{ }\text{2} .\text{2}\textrm{ }+\textrm{ }\text{2} .\text{4}\textrm{ }+\textrm{ }\text{2} .\text{6}\textrm{ }+\textrm{ }\text{2} .\text{8}\textrm{ }+\textrm{ }\text{3} .\text{0}\textrm{ }+\textrm{ }\text{3} .\text{2}\textrm{ }+\textrm{ }\text{3} .\text{4}$

$\text{13}\textrm{ }+\textrm{ }\text{21}\textrm{ }+\textrm{ }\text{29}\textrm{ }+\textrm{ } \ldots +\textrm{ }\text{69}$

$\sum_{k = 1}^{10} 5 - 6 k$

On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?

A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?

Use the formula to find the indicated partial sum of each geometric series.

1. ⓐ $S_{11}$ for the series $\text{8}\textrm{ }+\textrm{ }-\text{4}\textrm{ }+\textrm{ }\text{2}\textrm{ }+\textrm{ } \ldots$
2. ⓑ $\sum^{​}_{k = 1}^{6} 3 \cdot 2^{k}$

$S_{20}$ for the series $\text{1},\text{000}\textrm{ }+\textrm{ }\text{500}\textrm{ }+\textrm{ }\text{250}\textrm{ }+\textrm{ } \ldots$

$\sum_{k = 1}^{8} 3^{k}$

At a new job, an employee’s starting salary is $26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years.

At a new job, an employee’s starting salary is $32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years?

Determine whether the sum of each infinite series is defined.

1. ⓐ $\text{12}\textrm{ }+\textrm{ }\text{8}\textrm{ }+\textrm{ }\text{4}\textrm{ }+\textrm{ } \ldots$
2. ⓑ $\frac{3}{4} + \frac{1}{2} + \frac{1}{3} + ...$
3. ⓒ $\sum_{k = 1}^{\infty} 27 \cdot \left(\frac{1}{3}\right)^{k}$
4. ⓓ $\sum_{k = 1}^{\infty} 5 k$

$\frac{1}{3} + \frac{1}{2} + \frac{3}{4} + \frac{9}{8} + ...$

$24 + \left(−12\right) + 6 + \left(−3\right) + ...$

$\sum_{k = 1}^{\infty} 15 \cdot \left(– 0.3\right)^{k}$

Find the sum, if it exists, for the following:

1. ⓐ $10 + 9 + 8 + 7 + \ldots$
2. ⓑ $248.6 + 99.44 + 39.776 + \ldots$
3. ⓒ $\sum_{k = 1}^{\infty} 4 , 374 \cdot \left(– \frac{1}{3}\right)^{k – 1}$
4. ⓓ $\sum_{k = 1}^{\infty} \frac{1}{9} \cdot \left(\frac{4}{3}\right)^{k}$

$2 + \frac{2}{3} + \frac{2}{9} + ...$

$\sum_{k = 1}^{\infty} 0.76 k + 1$

$\sum_{k = 1}^{\infty} \left(- \frac{3}{8}\right)^{k}$

A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?

At the beginning of each month, $200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?

What is an $n \text{th}$ partial sum?

What is a geometric series?

What is an annuity?

Express each description of a sum using summation notation.

The sum from of $n = 0$ to $n = 4$ of $5 n$

Express each description of a sum using summation notation.

The sum that results from adding the number 4 five times

Express each arithmetic sum using summation notation.

$10 + 18 + 26 + \ldots + 162$

Use the formula for the sum of the first $n$ terms of each arithmetic sequence.

$\frac{3}{2} + 2 + \frac{5}{2} + 3 + \frac{7}{2}$

Use the formula for the sum of the first $n$ terms of each arithmetic sequence.

$3.2 + 3.4 + 3.6 + \ldots + 5.6$

Express each geometric sum using summation notation.

$8 + 4 + 2 + \ldots + 0.125$

Use the formula for the sum of the first $n$ terms of each geometric sequence, and then state the indicated sum.

$9 + 3 + 1 + \frac{1}{3} + \frac{1}{9}$

Use the formula for the sum of the first $n$ terms of each geometric sequence, and then state the indicated sum.

$\sum_{a = 1}^{11} 64 \cdot 0.2^{a - 1}$

Determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

$2 + 1.6 + 1.28 + 1.024 + ...$

Determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

$\sum^{​}_{\infty}^{k = 1} - \left(- \frac{1}{2}\right)^{k - 1}$

Use the geometric series $\sum_{k = 1}^{\infty} \left(\frac{1}{2}\right)^{k} .$

What number does $S_{n}$ seem to be approaching in the graph? Find the sum to explain why this makes sense.

Find the indicated sum.

$\sum_{n = 1}^{6} n \left(n - 2\right)$

Find the indicated sum.

$\sum_{k = 1}^{7} 2^{k}$

Use the formula for the sum of the first $n$ terms of an arithmetic series to find the sum.

$6 + \frac{15}{2} + 9 + \frac{21}{2} + 12 + \frac{27}{2} + 15$

Use the formula for the sum of the first $n$ terms of an arithmetic series to find the sum.

$\sum_{k = 1}^{11} \left(\frac{k}{2} - \frac{1}{2}\right)$

Use the formula for the sum of the first $n$ terms of a geometric series to find the partial sum.

$S_{7}$ for the series $0.4 - 2 + 10 - 50...$

Use the formula for the sum of the first $n$ terms of a geometric series to find the partial sum.

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

Find the sum of the infinite geometric series.

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

Find the sum of the infinite geometric series.

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

Determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

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

Determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

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

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

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

The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series?

Karl has two years to save $\$ 10 , 000$ 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?

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 pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels $\frac{3}{4}$ the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?