Tuesday, August 17, 2021 6:09 PM
Microsoft Word - MT315 Midterm.docx
MT 315 Midterm
1. A birdcage contains 4 macaws, 4 pa
ots, 4 canaries, and 4 finches. Each group has a bird of
the colors white, green, red, and yellow.
a. How many ways can a blue bird be chosen?
. How many ways can a green bird or a canary be chosen?
c. How many ways can three birds be chosen so that each have a different color?
2. Use the Binomial Theorem to expand (3? − 2)!
3. Find 2 − 6 + 18 −⋯+ 13122
4. Consider the sequence given by ?! = 2 − 6?.
a. Find the first 6 terms of the sequence. What sort of sequence is this?
. Find the sum of the first 55 terms. That is, compute ∑ ?"##"$% .
5. Use polynomial fitting to find a closed formula for the sequence (??)? ∈ ℕ0:
?% + ?& + 1 = 1, 3, 13, 37, 81,…
6. Consider the recu
ence relation ?' = ?'() − 2?'(& with first two terms ?0 = 0 and ?1 = 1.
a. Write out the first 5 terms of the sequence defined by this recu
ence relation.
. Solve the recu
ence relation. That is, find a closed formula for an.
7. Prove the following using induction.
-?&
!
"$%
= 1& + 2& + 3& +⋯+ ?& = XXXXXXXXXX +⋯+ ?)'
[Hint: Recall that XXXXXXXXXX +⋯+ ? = !(!)%)
'
]