Cards In This Deck

• $$\begin{gathered} \text{True or False?} \\ \text{A recurrence relation given by }{u}_{n+1}=u_n+3\text{ with }{u}_1=2\text{ generates a geometric sequence.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{This generates an arithmetic sequence because you are adding a constant (}d=3\text{) to each term.} \\ \text{A geometric recurrence relation would look like }{u}_{n+1}=3u_n\text{.} \end{gathered}$$
• $\text{Given two non-consecutive terms of an arithmetic sequence (e.g., }{u}_5\text{ and }{u}_{10}\text{), what is the standard method to find }a\text{ and }d\text{?}$
  $$\begin{gathered} \text{Set up two simultaneous equations using }{u}_n=a+(n-1)d\text{.} \\ \text{E.g., }a+4d={value}_1\text{ and }a+9d={value}_2\text{. Then subtract one from the other.} \end{gathered}$$
• $\text{What defines the order of a periodic sequence?}$
  $$\begin{gathered} \text{The smallest positive integer }k\text{ such that:} \\ {u}_{n+k}=u_n\text{ for all }n\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{The number of terms in the sum }{\sum }_{r=5}^{20}(3r+1)\text{ is }20-5=15\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Count: }20-5+1=16 \\ \text{It is (Top Limit) - (Bottom Limit) + 1.} \end{gathered}$$
• $$\begin{gathered} \text{What is the first step to prove the sum of an arithmetic series formula:} \\ {S}_n=\frac{n}{2}(2a+(n-1)d)\text{?} \end{gathered}$$
  $\text{Write the sum out twice: once in the correct order and once in reverse order, then add the two equations together.}$
• $\text{When modelling a real-life situation (e.g., salary increase), how do you decide between an arithmetic and geometric model?}$
  $$\begin{gathered} \text{Arithmetic: Increases by a fixed amount (e.g., +}500/year). \\ Geometric:Increasesbyapercentage/proportion(e.g.,+5 \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{If a problem asks for the *least number of terms* }n\text{ for a sum to exceed a value, and the calculation gives }n>15.2\text{, the answer is }15\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: 16} \\ n\text{ must be an integer. If }n\text{ must be greater than 15.2, the first integer satisfying this is 16.} \end{gathered}$$
• $\text{How do you calculate the sum of a series from term }k\text{ to term }n\text{ (i.e., }{\sum }_{r=k}^n{u}_r\text{) using the sum formula }{S}_n\text{?}$
  ${\sum }_{r=k}^n{u}_r=S_n-S_{k-1}$
• $\text{When calculating the sum of a geometric series, when is it better to use }{S}_n=\frac{a(r^n-1)}{r-1}\text{ instead of }{S}_n=\frac{a(1-r^n)}{1-r}\text{?}$
  $\text{When }r>1\text{.}$
• $\text{What is the specific condition for a geometric series to converge (have a sum to infinity)?}$
  $$\begin{gathered} |r|<1 \\ \text{(Or equivalently: }-1<r<1\text{)} \end{gathered}$$
• $$\begin{gathered} \text{What is the first step to prove the sum of a geometric series formula:} \\ {S}_n=\frac{a(1-r^n)}{1-r}\text{?} \end{gathered}$$
  $\text{Write down the expression for }{S}_n\text{, then multiply the entire equation by }r\text{ to get an expression for }r{S}_n\text{. Then subtract the two equations (}{S}_n-r{S}_n\text{).}$
• $$\begin{gathered} \text{True or False?} \\ \text{The sequence }5,-10,20,-40,\dots \text{ is a decreasing sequence.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{This is an alternating sequence (or oscillating). A decreasing sequence requires }{u}_{n+1}<u_n\text{ for all }n\text{. Here, }-10<5\text{ but }20>-10\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{The 10th term of a geometric sequence is given by:} \\ {u}_{10}=a{r}^{10} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Formula: }{u}_n=a{r}^{n-1} \\ \text{So, }{u}_{10}=a{r}^9 \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{When solving for }n\text{ in the inequality }0.8^n<0.05\text{, the inequality sign flips when you take logarithms.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False (at the log step)} \\ \text{Taking logs (e.g., }\ln (0.8^n)<\ln (0.05)\text{) does not flip the sign. } \\ \text{However, the sign WILL flip when you divide by }\ln (0.8)\text{ because }\ln (0.8)\text{ is a negative number.} \end{gathered}$$