IB Mathematics HL
Year I—Unit 3 Test
Date:
Name:
Period:
Part 1, no calculators, 50 points.
1. (6 points) You are given an arithmetic sequence u1 , u2 , . . . with
u4 = 15 and
u8 = 7. Determine
(a) u1 ,
(b) d (the common difference), and
(c) the sum S22 =
22
X
uk .
k=1
2. (6 points) Find the average of the positive terms of the arithmetic
sequence beginning with 92 and having common difference −7.
3. (6 points) The sum of an infinite geometric sequence is 13 21 , and
the sum of the first three terms is 13. Find the first term.
4. (5 points) Determine the term independent of x in the expansion
!
1 8
3
of 2x −
.
x
5. (5 points) You are given that
(1 − 2x)n = 1 − 12x + px2 + qx3 + · · · .
Find p, q, and n.
6. (5 points) Suppose that we have a big jar of 20 marbles. Of these
marbles, 8 are red, 6 are blue, 4 are black, and 2 are white. In how
many ways can you reach in and grab 5 marbles such that exactly
3 of them be blue, and the other two are of different colors?
7. (5 points) In how many ways can your class of 13 students be
divided into three distinct groups of 4, 5, and 3 students?
8. (5 points) You are given the factorization
16, 170 = 2 × 3 × 5 × 72 × 11.
Compute the number of divisors of 16,170.
9. (3 points) Let n be a positive integer and compute
n
X
Ñ é
(−2)k
k=0
n
.
k
10. (4 points) How many distinct rearrangements are there of the
“word” INABILITY?
IB Mathematics HL
Year I—Unit 3 Test
Date:
Name:
Period:
Part 2, calculators allowed, 50 points.
1. (6 points) Compute the coefficient of x3 in the expansion of
(2 − x)(3 + 2x)10 .
2. You are given an alternating geometric series such that a1 = 10
and a3 = .4.
(a) (4 points) Compute Sn =
n−1
X
ak in terms of n.
k=0
(b) (3 points) Compute S∞ =
∞
X
ak .
k=0
(c) (4 points) Find the least value of n making |Sn − S∞ | < 10−6 .
3. You are given the first few terms of a sequence as follows:
3, 8, 15, 24, 35, 48.
(a) (3 points) Using the idea of higher-order differences determine the nature of this sequence (arithmetic, quadratic, cubic,
and so on).
(b) (6 points) Determine a formula for the n-th term, un of this
sequence.
(c) (6 points) You are given the formulas
n
P
k=1
k2 =
1
6 n(n
+ 1)(2n + 1),
n
P
k=1
k3 =
1 2
4 n (n
+ 1)2 .
Using this, together with your answer to part (b) above, comn
P
pute
uk as a polynomial in n.
k=1
4. (6 points) How many four-digit numbers are there which contain
at least one digit 3?
(Note: a four-digit number cannot begin with 0.)
5. (6 points) The SAS (Puxi) mathematics department consists of
faculty with names Burke, LeCoq, Michel, Leipold, Russell, Surowski,
Wager, Yuen, and Zhu. In how many ways can these faculty
members be chosen to sit on the stage in four chairs, aligned in
a row, in such a way that Surowski and Zhu cannot sit next to
each other?
6. (6 points) Compute 23 + 25 + 92 +
2
25
+
2
27
+
2
125
+ ···.