October 21, 2009
1. Expand step-by-step: (4a-3b)7
a. What are the coefficients?
7C0 7C1 7C2
b. Expand (x+y)7
c. Substitute
7C3
7C4
7C5
7C6
7C7
7y0 + C x6y1 + C x5y2+ C x4y3 +
C
x
7 0
7 1
7 2
7 3
3 4
2 5
1 6
0 7
7 C4 x y + 7 C5 x y + 7 C6 x y + 7 C7 x y
1(4a)7(-3b)0 + 7(4a)6(-3b)1 + 21(4a)5(-3b)2+
35(4a)4(-3b)3 + 35(4a)3(-3b)4 + 21(4a)2(-3b)5
d. Simplify
+ 7(4a)1(-3b)6 + 1(4a)0(-3b)7
16384a7- 86016a6b + 193536a5b2 - 241920a4b3 +
3 4
2 5
6
7
181440a b - 81648a b +20412ab - 2187b
October 21, 2009
2.How many five character code words are
there if the first character is always a letter
and the other characters are letters/ and or
digits?
26*36*36*36*36 = 43,670,016
How many if letters or numbers can NOT
be repeated?
26*35*34*33*32 = 32,672,640
October 21, 2009
3. A club has 45 members, and its
membership committee has three
members.
a) How many different membership
committees are possible?
C3 = 14190 (Order doesn't matter)
45
b) What if the three positions are different?
How many are possible?
45P3
= 85140
October 21, 2009
4. State whether the given sequence if arithmetic,
geometric, or neither.
If arithmetic, give the common
difference and if geometric, the common ratio.
a.27,-18,12,-8,...
Geometric, r = -2/3
b.2,5,10,17,...
Neither
c. 3,8,13,18,...
Arithmetic, d = 5
October 21, 2009
5. The sequence is arithmetic.-4,1,6,11,...
a) Find the common difference
d=5
b) Find the tenth term.
tn = -4 + (n - 1)5
t10 = -4 + (10 - 1)5 = 41
c) Find the Formula for the nth term.
tn = -4 + (n - 1)5
October 21, 2009
6. The sequence is geometric.3,6,12,24,...
a) Find the common ratio
r = 6/3 = 12/6 = 24/12 = 2
b) Find the tenth term.
t10 = 3(2)(10-1) = 1536
c) Find the Formula for the nth term.
t10 = 3(2)(n-1)
October 21, 2009
7. Give a recursive definition of the
sequence: 1,2,4,7,11,...
t1 = 1
tn = tn-1 + ??
October 21, 2009
Find the missing terms of an arithmetic
sequence whose first term is -20 and sixth
term is 370.
tn = -20 + (n-1)d
t6 = 370 = -20 + (6 - 1) d
390 = 5d
78 = 3
tn = -20 + (n-1)78
October 21, 2009
Find the missing terms of a geometric
sequence whose first term is -1/81 and
and seventh term is -9.
tn = -1/81(r)(n-1)
t7 = -9 = -1/81(r)(7-1)
729 = r6
+/- 3 = r
tn = -1/81(3)(n-1)
or
tn = -1/81(-3)(n-1)
October 21, 2009
Write the series in sigma notation:
-2 + (-5) + (-8) + ...+ (-20)
7
!
n=1
(-2 + (n -1)(-3) )
October 21, 2009
Find the sum of the arithmetic series if it exists.
a)
Total Terms = (28-14) + 1 = 15
-22 + -20 + -18 + ..... + 6
S15 = (15/2)(-22 +6) = -120
b) 2+4+6+8+...
No finite sum, infinite series....
c) 5+7+9+11+...+27
tn = 5 + (n - 1)2
27 = 5 + (n - 1)2
22 = (n - 1)2
11 = (n -1)
12 = n
S12 = 12/2(5 + 27)
S12 = 6(32) = 192
October 21, 2009
Find the sum of the geometric series if it
exists.
a)
5 Terms t1 = 12 r = 3
S5 = 12 ( 1 - 35)/(1-3) = 1452
b) 4+ 4/3 + 4/9 + 4/27 +.....
tn = 4 (1/3)(n-1) Infinite series, |r|<1, therefore it converges
S = t1/(1 - r) = 4/(1-1/3) = 6
c) 1/48 + 1/16+ 3/16 + 9/16 + ...
tn = 1/48(3)(n-1) |r| > 1, therefore it diverges, no finite sum