**1.** Write a rule for the nth term of the arithmetic sequence with a1= -7 and the common difference of 5/2.

**2.** Find the sum of the first 10 terms of the arithmetic series. -5 + 0 + 5 + 10 +...

* a*.350

3. Find the sum of the first 22 terms of the arithmetic sequence, if the first term is –2 and the common difference is –5.

__a.__–1199

4. Find the common difference of the arithmetic sequence.

0, 0.4, 0.8, 1.2, . . .

** **__ a.__–0.3

5. Tell whether the sequence is arithmetic. If it finds the common difference.

a. -5, -1, 3, 7, 11,...

b. -1, -1/3, 1/3, 1 , 5/3,...

6. Identify the sequence as __arithmetic, geometric, or neither.__

1, 1, 2, 3, 5, 8, 13, . . .

Mathrules
Jul 5, 2018

#1**0 **

Arithmetic sequences are written in the form of the following:

\(a_n=a_1+d(n-1)\).

a_{n} is the nth term of the sequence

a_{1} is the first term of the sequence

d is the common difference

n is the desired term number

1) We already know the information necessary to write an equation for the nth term of this sequence.

\(a_1=-7; d=\frac{5}{2}\\ a_n=-7+\frac{5}{2}(n-1)\) | The only thing left to do is simplify. Distributing is the first step to accomplish this. |

\(a_n=-\frac{14}{2}+\frac{5}{2}n-\frac{5}{2}\) | Combine like terms. |

\(a_n=\frac{5}{2}n-\frac{19}{2}\) | This is completely simplified. |

2) There is a formula that exists for the summation of an arithmetic series or geometric series, but that probably would not help your understanding anyway; I can derive it for you, though.

We have to know the last term, and we can generate this by using the formula from before: \(a_n=a_1+d(n-1)\). Therefore, n=10, d=5, and a_{1}=5:

\(a_{10}=-5+5(10-1)\) | Evaluate this to determine the last term of the sequence. |

\(a_{10}=-5+5*9\) | Simplify the right hand side. |

\(a_{10}=40\) | |

Now, let's attempt to evaluate the sum. Standard notation dictates \(S_n\) for the summation.

\(S_{10}=-5+0+...+35+40\\ S_{10}=\hspace{2mm}40+35+...+0\hspace{2mm}-5\\ \overline{\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad}\\ 2S_{10}=\underbrace{35+35+...+35+35}=35*10=350\\ \hspace{24mm}\text{10 times}\\ S_{10}=175\) | All I did here is reverse the same sum and added both of them together. This made it significantly easier to determine the sum. |

I did not answer every question here because the others can be answered with the knowledge given above, albeit not directly. Try and figure it out yourself.

TheXSquaredFactor
Jul 6, 2018