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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?

 Dec 23, 2020
 #1
avatar+25658 
+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?

 

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

 

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

 

laugh

 Dec 23, 2020
 #3
avatar+117725 
+1

Nice,heureka  !!!!

 

 

cool cool cool

CPhill  Dec 23, 2020
 #4
avatar+25658 
0

Thank you, CPhill ! 

 

laugh

heureka  Dec 24, 2020
 #2
avatar+117725 
+1

1/1300  =  1/13 *  ( 1/100)  =  .00076923

 

The part in red is the repeating part with a period of 6

 

So

 

When  n  = 16

 

2 + 6 (16)  =  2 + 96  = 98 digits  to the right of the decimal point

 

So....the 100th digit will be the second digit in the period  =  7

 

cool cool cool

 Dec 23, 2020

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