# Sequences and Series

## Self-Test:

 1)       Consider the sequence $-1,-3, -5,-7, -9, \ldots$. The formula for $a_{n}$ is given by: $a_{n} = (-1)^{n} (2n-1)$ $a_{n} = 2n+1$ $a_{n} = 2n - 1$ $a_{n} = -2n+1$ $a_{n} = (-1)^n(-2n+1)$

 2)       Consider the sequence $1, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{4}, \frac{1}{5}, -\frac{1}{6}, \ldots$. Determine which of the following expressions gives a formula for $a_{n}$. $a_{n} = \frac{1}{n}$ $a_{n} =\frac{-1}{n}$ $a_{n} = \frac{(-1)^{n}}{n}$ $a_{n} = \frac{(-1)^{n+1}}{n}$ $a_{n} =( \frac{-1}{n})^{n+1}$

Hint Notice that the sign keeps alternating. Be mindful of whether it's positive for $n$ even, or for $n$ odd.

 3)       If $a_{1} = 1$, and $a_{n} = 2a_{n-1} +1$, then $a_{7} =$? $15$ $127$ $63$ $257$ None of the Above

Hint List all the terms $a_{1}, a_{2}, \ldots a_{7}$ using the recursion formula.

 4)       Consider the sequence $1, 1, 4, 10, 28, 76, \ldots$. Then $a_{n}$ is given by: $a_{1} = 1, a_{2} = 1, a_{n} = 2(a_{n-1} + a_{n-2})$ $a_{1} = 1, a_{2} = 1, a_{n} = 2a_{n-1} + a_{n-2}$ $a_{1} = 1, a_{2} = 1, a_{n} = a_{n-1} + a_{n-2}$ $a_{1} = 1, a_{2} = 1, a_{n} = 2^{a_{n-1} + a_{n-2}}$ None of the above

Hint Which ones fit the pattern?

 5)       Consider the statement Consider the sequence $3, 14, 25, 36, 47,\ldots$. The formula for $a_{n}$ is given by: $a_{n} = 3+11n$ $a_{n} = 3 \cdot 11^{n}$ $a_{n} = \frac{1}{24} \big(23 n^4-234 n^3+829 n^2-930 n+648\big)$ $a_{n} = 11 + 3(n-1)$ None of the above

Hint What type of sequence is this? Geometric? Arithmetic?

 6)       The sequence $3, 31, 314, 3142, 31425, \ldots$ is: arithmetic geometric recursive None of the Above

Hint Check for difference, ratio, relation between previous terms.

 7)       Find the sum $\frac{2}{3} + \frac{2}{9} + \cdots +\frac{2}{729}$. $\frac{364}{243}$ $\frac{2}{3}$ $1$ $\frac{728}{729}$ None of the Above

Hint What type of series is this? Arithmetic? Geometric?

 8)       If we write the series $1-\frac{2}{5} + \frac{3}{25} - \frac{4}{125} + \frac{5}{625} - \cdots$ in sigma notation we get: $\sum_{i=0}^{\infty} \frac{ (-1)^{n+1} n}{5^{n}}$ $\sum_{i=0}^{\infty} \frac{n}{5^{n}}$ $\sum_{i=0}^{\infty} (\frac{-n}{5})^{n}$ $\sum_{i=0}^{\infty} \frac{n}{-5^{n}}$ None of the Above

Hint Be mindful of the sign!

 9)       The series $\sum_{i=1}^{\infty} \frac{3^{i}}{4^{i+1}}$ is equal to: $1$ $\frac{1}{4}$ $\frac{3}{4}$ $\frac{1}{3}$ None of the Above

Hint You might need to start by re-indexing.

 10)       Consider a geometric sequence where the fourth term, $t_4 = 54$ and the sixth term, $t_6 = 486$. The common ratio of this sequence is: $-3$. $3$ $89$ $216$ No such sequence exists

Hint Note that these are not successive terms!