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Page 443: Further Your Understanding
$t_n=t_{n-1}+t_{n-2}$
But they differ only in their first 2 terms. Here, we will make our own sequence by changing the first two terms.
We will choose $t_1=2$, $t_2=3$
$t_3=2+3=5$
$t_4=3+5=8$
$t_5=5+8=13$
$t_6=8+13=21$
$t_7=13+21=34$
$t_8=21+34=55$
$dfrac{t_4}{t_3}=dfrac{8}{5}=1.6$
$dfrac{t_5}{t_4}=dfrac{13}{8}=1.625$
$dfrac{t_6}{t_5}=dfrac{21}{13}=1.6154$
$dfrac{t_7}{t_6}=dfrac{34}{21}=1.6159$
$dfrac{t_8}{t_7}=dfrac{55}{34}=1.6176$
Notice that ratios tend to approach $1.618approx dfrac{1+sqrt{5}}{2}$
This is the same ratio as that of Fibonacci and Lucas sequences.
$t_n=t_{n-1}+t_{n-2}$
Notice that for a geometric sequence with common ratio $r$, the terms are
$a,;ar^2,;ar^3,;ar^4$
Combining with the recursive formula for Fibonacci sequence
$t_n=t_{n-1}+t_{n-2}$
We shall have
$ar^n=ar^{n-1}+ar^{n-2}$
For $n=2$
$r^2=r+1$
$r^2-r+1=0$
This is a quadratic equation with $a=1$, $b=-1$, and $c=1$
which we shall solve using quadratic formula
$r=dfrac{1pmsqrt{(-1)^2-4(1)(1)}}{2(1)}=dfrac{-1pmsqrt{5}}{2}$
The value of the positive root, is $approx 1.618$ which is the common ratios of consecutive terms for Fibonacci and Lucas sequences.
1.618
$$
$t_1=1$
$t_2=5$
$t_3=5+2(1)=7$
$t_4=7+2(5)=17$
$t_5=17+2(7)=31$
$t_6=31+2(17)=65$
$t_7=65+2(31)=127$
$t_8=127+2(65)=257$
$t_9=257+2(127)=511$
$t_10=511+2(257)=1025$
$dfrac{t_2}{t_1}=5$
$dfrac{t_3}{t_2}=dfrac{7}{5}=1.4$
$dfrac{t_4}{t_3}=dfrac{17}{7}=2.4286$
$dfrac{t_5}{t_4}=dfrac{31}{17}=1.8235$
$dfrac{t_6}{t_5}=dfrac{65}{31}=2.0968$
$dfrac{t_7}{t_6}=dfrac{127}{65}=1.9538$
$dfrac{t_8}{t_7}=dfrac{257}{127}=2.0236$
$dfrac{t_9}{t_8}=dfrac{511}{257}=1.9883$
$dfrac{t_{10}}{t_9}=dfrac{1025}{511}=2.0059$
The ratios tend to approach 2.
If $t_n=2^n$, then we should have the following sequence
$2,;4,;8,;16,;32,;64,;128,;256,;512;1024$
but the actual sequence is
$1,;5,;7,;17,;31,;65,;127,;257,;511,;1025$
Observe that if $n$ is odd, $t_n=2^n-1$ and if $n$ is even, $t_n=2^n+1$
Therefore, we can use $(-1)^n$ to incorporate this behavior resulting in
$t_n=2^n+(-1)^n$
b) 5, 1.4, 2.43, 1.82, 2.10, 1.95, 2.02, 1.99, 2.00 , it gets closer to 2
c) $t_n=2^n+(-1)^n$