1. A sequence is formed by adding $2$ to the triple of the previous term. If the first term is $1$, how many multiples of $6$ less than $10{,}000$ would be terms of the sequence?

2. Recall that a Fibonacci sequence is one in which all entries from the third onward are the sum of the two entries before it. If the $5$th and $8$th entries of a Fibonacci Sequence are $19$ and $79$, respectively, then what is the first entry of this sequence?

3. We know that $\dfrac{1}{13}=0.0769230769230769230769230769230769\cdots$ When $\dfrac{1}{1300}$ is written as a repeating decimal, what is its $100$th digit to the right of the decimal point?

Guest Dec 23, 2020

#1**+2 **

**2. **

**Recall that a Fibonacci sequence is one in which all entries from the third onward are the sum of the two entries before it. If the 5th and 8th entries of a Fibonacci Sequence are 19 and 79, respectively, then what is the first entry of this sequence?**

\(\begin{array}{|lrcll|} \hline & a_5 &=& 19 \\ & a_8 &=& 79 \\ \hline &\mathbf{ a_6+a_7} &=& \mathbf{a_8} \\ (1) & a_6+a_7 &=& 79 \\ \hline &\mathbf{ a_5+a_6} &=& \mathbf{a_7} \\ (2)& 19+a_6 &=& a_7 \\ \hline (1)+(2): & 19+2a_6+a_7 &=& 79+a_7 \\ &19 + 2a_6 &=& 79 \\ & 2a_6 &=& 79-19 \\ &2a_6 &=& 60 \quad &| \quad : 2 \\ & \mathbf{a_6} &=& \mathbf{30} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline a_4+a_5 &=& a_6 \\ a_4 + 19 &=& 30 \\ \mathbf{a_4} &=& \mathbf{11} \\ \hline a_3+a_4 &=& a_5 \\ a_3 + 11 &=& 19 \\ \mathbf{a_3} &=& \mathbf{8} \\ \hline a_2+a_3 &=& a_4 \\ a_2 + 8 &=& 11 \\ \mathbf{a_2} &=& \mathbf{3} \\ \hline a_1+a_2 &=& a_3 \\ a_1 + 3 &=& 8 \\ \mathbf{a_1} &=& \mathbf{5} \\ \hline \end{array}\)

heureka Dec 23, 2020