Test #4 Review

MATH 1314 – TEST #4 REVIEW – CHAPTERS 6 & 8

x

 2

y

 3

z

  5 1. Write the augmented matrix for the following system: 2. Solve using Gaussian Elimination (ROWOPS): 

x

 2

y

 3

z

  5 a)   5 6

x x

 4 5

y y

 2 1 b)   2

x y z x y z

8 1   2

x y z x y z

c)  8

x

1  2     2 3

x x y

5

z

2 1 0 3. Suppose when solving a system using Gaussian Elimination, you ended up with the following matrix. Write the solution. a)     1 0 0 1 0 0  7 0  5 4 0 0     b)     1 0 0 1 0 0  1 2 0 3 6  4     4. Solve using back-substitution:     1 5 0 1 0 0 5. Write the first four terms of each sequence: a) a n = 7n – 4 b) a n = 1 (

n

 1)!

 2 1 3 10   1 2   6. Find each sum: a)

i

5   1  2

i

2  3  b)

i

4   1  

i

 1 

i

!

7. Write the first 4 terms of each arithmetic sequence: a) a 1 = 7, d = 4 b) a 1 = –4, d = –5

8. Find the indicated term of the arithmetic sequence (use a formula!) a) Find a 6 when a 1 = 5, d = 3 b) Find a 12 when a 1 = –8, d = –2 c) Find a 14 when a 1 = 14, d = –4 9. Write a formula for the general n th term of each arithmetic sequence, then find a 20 . a) –7, –3, 1, 5, … b) 200, 180, 160, … 10. a) Find the sum of the first 22 terms of the arithmetic sequence 5, 12, 19, 26, … b) Find the sum of the first 15 terms of the arithmetic sequence –6, –3, 0, 3, … 11. A company offers a starting salary of \$31,500 with raises of \$2300 per year. Find the total salary over a ten-year period. 12. A theater has 25 seats in the first row and 35 rows in all. Each successive row contains one additional seat. How many seats are in the theater? 13. Write the first 4 terms of each geometric sequence: a) a 1 = 3, r = 2 b) a 1 = 16, r = ½ 14. Find the indicated term of the geometric sequence (use a formula!) a) Find a 7 when a 1 = 2, r = 3 b) Find a 10 when a 1 = 16, r = ½ c) Find a 5 when a 1 = –3, r = 2 15. Write a formula for the general n th term of each geometric sequence, then find a 8 . a) 1, 2, 4, 8, … b) 100, 10, 1, 1 / 10 , … 16. a) Find the sum of the first 15 terms of the arithmetic sequence 5, 15, 45, 135, … b) Find the sum of the first 7 terms of the arithmetic sequence 8, 4, 2, 1, … 17. A job pays \$32,000 for the first year with an annual increase of 6% per year. a) What is the salary in the sixth year? (round to the nearest dollar) b) What is the total salary paid over the 6 years? (round to the nearest dollar) 18. Find the sum of each infinite series: a)

9 3 1

b)

i

   1  

i

 1 c)

i

   1

i

 1

1.  1 2   2 1 1 1 3 1  1  5 1   8   2. a)   5 6  1  0   4 2 5 1    5 4  1 5  1 0 6  0 1 7   2 5  7 5   1 5 4 5

R

1  6

R

1  5

R

2

R

2  

R Solution R

2 1 4. 5.  1 2    1 2 3    2 1 1 1 1  1 8  5 1     2 1

R R

1 1 0 0  3  1 3   5 5 11    4 13    3 1 1

R R

2 2   2

R

2

R R

3 2  

R

3

R

1 2. b)  1 0   0 1 0 0  1 3 5 3  7 3 7 3  11 3 28 3      3 7

R

3 1 3  5 3

R

3

R

3  

R

1

R

2  1 0 0 1   0 1 0 3 0 0 1  4    

Solution

 3. a) Infinite Solutions  b) No Solution (–83, 16, 2) a) 3, 10, 17, 24 b) 1 1 2 6 8. 10.   a) a 6 = 5 + (6 – 1)(3) =

20

b) a 12 = –8 + (12 – 1)(–2) =

–30

c) a 14 = 14 + (14 – 1)(–4) =

–38

a) a 22 = 5 + (22 – 1)(7) = 152  S 22 = b) a 15 = –6 + (15 – 1)(3) = 36  S 15 = 22 2 15 2    1    3 1  2  2 1 0 0  1   0 5  2    1    0   0 4 2 1 0 5 3 3   1 2     2. c)  1 0    0 1 0 0  1 5  3 5 27 5 2 5 1 5  14 5     3

R

1  2

R

1 

R

2 

R

3 2  5 1

R R

2 2   4

R

2

R

1 

R

3 1 5 3 5 5 27

R R

3 3 

R

3 

R

2

R

1  1 0 0    0 1 0 0 0 1 8 27  1 9  14 27    

Solution

:  8 27 9.  

= 1727 = 225

 1 7. a) 7, 11, 15, 19 b) –4, –9, –14, –19 a) a n = 4n – 11 ; 69 b) a n = 220 – 20n; –180  14 27 

11. 12. 13. 14. 15. a 1 = 31,500, d = 2,300 a 10 = 31500 + (10 – 1)(2300) = 52200  S 10 = 10 2  a 1 = 25, d = 1 a 35 = 25 + (35 – 1)(1) = 59  S 35 = a) 3, 6, 12, 24 b) 16, 8, 4, 2 a) a 7 =   7 1 =

1458

b) a 10 = 16   10 1 =

1

/

32

c) a 5 =    5 1 =

–48

a) a n = 1(2)

n

 1 ; 128 b) a n = 100

n

 1 ; 1 100,000 16. a) S 15 = 5(1 3 ) (1 3)

= 35,872,265

b) S 7 = 8(1 (1   1 2 7 1 2 ) )

127

8

or 15.875

17. a) a b) S 6 6 = = 32000(1.06) = 6

35 2  

= \$418,500

18. a)

S

 b)

S

   (1 9  1 3 )

27

2

5

c)