Pendulum sequences. Start with a negative number, such as $-9.$ Write it down. Choose a positive

File Preview uch as $2,$ and write it next to the first number. Continue the line of numbers, with each

following number being the sum of the two previous numbers, until the next term would break the

alternating negative $/$ positive pattern. In this case, we would get $-9,2,-7$ and then we would not

be able to continue because $2+(-7)=-5,$ which has the same sign as $-7.$ The goal is to prolong

the pendulum sequence, swinging back and forth between negatives and positives as long as possible.

You cannot include $0$ in the sequence.

$($a$)$ Starting with $-9,$ make a pendulum sequence with a length of at least $5.$

$($b$)$ Starting with $-9,$ make a pendulum sequence with a length of at least $10.$

$($c$)$ Starting with $-100,$ make a pendulum sequence with a length of at least $10.$

$($d$)$ Suppose that only integers may be used. Find two starting integers which will make a pendulum

sequence of length at least $10.$