Sequences and summations
MATH 174
1. Give the first five terms of the sequence whose closed-form expression is given. Tell if the
sequence is arithmetic, geometric, or neither.
(a) an = 7 − 2n where n = 0, 1, 2, . . .
(b) xj = (−1)j 3j where j = 0, 1, 2, . . .
(c) cm = m! where m = 0, 1, 2, . . .
2. Find a closed-form expression for each sequence whose recurrence relation is given. Then,
list the first five terms of that sequence and tell if the sequence is arithmetic, geometric, or
neither.
(a) bn = bn−1 + 4, b0 = 1
(b) An = −An−1 , A0 = −2
(c) xm =
1
xm−1 , x0 = 1
2
(d) yk = kyk−1 , y0 = 1
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3. Given the first few terms of a sequence, write a closed-form expression for that sequence.
(a) 1, −2, 4, −8, 16, . . .
393
(b) 49, 6, − 17 , − 50
49 , − 343 , . . .
4. Compute each sum.
(a)
5
X
k2
k=1
(b)
4
X
(−1)n
n=2
n
n−1
(c) Let p(a) = 0.1, p(b) = 0.6, and p(c) = 0.3. Compute
X
p(x).
x∈{a,b,c}
(d)
3 X
3
X
jk
k=1 j=1
(e)
9 k
X
1
k=1
2
Hint: 210 = 1024.
5. You take out a $5,000 loan to buy a fancy computer that plays Minecraft good. The
loan accrues interest monthly at an annual rate of 5%. Give a recurrence relation for the
amount you owe from month to month, assuming you make a monthly payment of $250.