Math 141 Exam 4: 8.2-9.5 Name:________________________ SCIENTIFIC CALCULATORS ONLY! YOU MUST SHOW WORK AS DONE IN CLASS VIDEO INSTRUCTION FOR CREDIT!!!
1. [6 pts] Solve for . 2. [6 points] Find the determinant of the following Matrix.
! ? 1 1 2 1 4 −1 4 3
! = 3 ) 1 3 −2 6 1 −5 8 2 3
–
3. [10 pts] Solve . ?! − 4?? + 3?! = 0
?! + ?? = 12
x
4. [20 pts] Find the inverse of the matrix using Gauss-Jordan elimination BY HAND.
) 1 −1 −1 0 2 1 3 2 0
–
Extra Credit: Given the system: ?” − 3?! = −2
2?” − 5?! = 1
a) Write a matrix equation in the form of ?? = ?.
b) Find ?#”.
c) Use ?#” to solve the system.
5. [8 pts each] Refer to the following matrices below:
? = #2 −1 3 0 4 −2
*;? = #−3 1 2 5
*;? = / −1 0 2 4 −3 1 −2 3 5
0;? = / 3 −2 0 −1 1 2
0
Find the following: a) B+AD b) DB 6. [3 pts] Write down the first 3 terms of the sequence.
{?$} = T (−1)$
(? + 6)(? + 3) W
7. [6 pts] Express the sum using summation notation.
1 2ln2
− 1
3ln3 +
1 4ln4
− 1
5ln5 + ⋯+
1 100ln100
8. [12 pts] Write the partial fraction decomposition of !”
#%!$%
9. [8 pts] Find the sum of ∑ ?&’%()* 10. [4 each] Given: 9 + 12 + 16 + !”
# + ⋯
a) Determine whether the following series is arithmetic b) Find the sum of the series. or geometric. If arithmetic, find its common difference,
, and if geometric, find its common ratio, .
d r
11. [8 pts] The first term of the geometric series is 3 and the third term is ” # . Find the fifth term.
12. [8 pts] Find the nth term of the arithmetic sequence given that the first term ?” = 6, and common difference ? =
$ %
Find the 53rd term: ?&’.
13. [7 pts] Expand Z1 − ” !( [ &
using the Binomial Theorem. Simplify completely. (Make sure you give the set-up)
14. [12 pts] Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. (Must show work as done in class video instruction.) 2′ + 4′ + 6′ + ⋯(2?)’ = 2?!(? + 1)!
