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# Sum Notation and Properties of Sigma

## Identify and state the sum of terms in finite series

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Sum Notation and Series Sums

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### Vocabulary

##### Complete the chart.
 Word Definition ________________ the large, stylized capitol E that means, approximately, "the sum of" Index __________________________________________________________ ________________ the expression being summed in a sigma Limits __________________________________________________________ ________________ when a series has a limit, and the limit exists ________________ when a series does not have a limit, or the limit is infinity Hypothesis __________________________________________________________ Mathematical induction __________________________________________________________ Partial sum __________________________________________________________

### Properties of Sigma

Label the sigma, index, summand, and limit:

\begin{align*}\sum_{n=1}^4 3n\end{align*}

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The above can be read as "find the _____ of the first ______ terms of the series, where the nth term is ____.

What does the sum above add to? __________

True or false: You can factor a coefficient out of a sum. ____________

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Express the Sum using Sigma Notation:

1. \begin{align*}1 + 5 + 9 + 13 + 17\end{align*}
2. \begin{align*}1 + \frac{1} {2} + \frac{1} {3} + \frac{1} {4} + ... + \frac{1} {10}\end{align*}

Find the series of numbers indicated and evaluate the summations:

1. \begin{align*}\sum_{n=-10}^{5} 7 -\frac{4}{3}(n-1)\end{align*}
2. \begin{align*}\sum_{n=-2}^{3} 8 -2(n-1)\end{align*}
3. \begin{align*}\sum_{n=-5}^{1} 5 +\frac{4}{3}(n-1)\end{align*}
4. \begin{align*}\sum_{n=1}^{6} 4(\frac{1}{2})^{n-1}\end{align*}
5. \begin{align*}\sum_{n=1}^{7} 3(-\frac{1}{2})^{n-1}\end{align*}
6. \begin{align*}\sum_{n=1}^{11} -3(\frac{4}{3})^{n-1}\end{align*}

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#### Gauss' Formula

Gauss' Formula allows us to add together the first positive integers:

\begin{align*}\sum = \frac{(n) (n + 1)} {2}\end{align*}

.In your own words, describe how Gauss' Formula works. _________________________________________________________________________

Explain how to use your calculator to solve \begin{align*}\sum_{n = 1}^9 n^2\end{align*} :

_________________________________________________________________________

Keep in mind Gauss' Formula can also be written as \begin{align*}S_n = \frac{n(a_1 + a_n)} {2}\end{align*}.

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Calculate the sums of the given series. You may use addition of individual terms or a series sum formula.

1. \begin{align*}\sum_{n = 0}^10 14 - \frac{1}{2}(n - 1)\end{align*}
2. \begin{align*}\frac{-71}{3} - \frac{67}{3} - 21 + ... + \frac{37}{3}\end{align*}
3. \begin{align*}2 + 4 + 6 + ... 26\end{align*}
4. \begin{align*}-2 - 1 + 0 + ... + 12\end{align*}
5. The first eight numbers of an arithmetic sequence add up to 604. The next eight numbers added up equal 156. Find the first number and the common difference in the sequence.
6. The first number in an arithmetic sequence is 80. Find the common difference if we also know that \begin{align*}s_9\end{align*} is eighteen times \begin{align*}a_{11}\end{align*}
7. If \begin{align*}a_n\end{align*} is an arithmetic sequence with \begin{align*} a_1 = 1\end{align*} . Find the second number if we know that the sum of the first five numbers is one-fourth of the sum of the next five numbers.
8. Given \begin{align*}(a_n) = 78, 75, 72, 69... \end{align*} Find \begin{align*}a_{150}\end{align*} and \begin{align*}s_{150}\end{align*}

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