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Pre-Calculus Pre-AP
Arithmetic, Geometric, and Recursively Defined Sequences
Work all problems on notebook paper.
Give recursive definitions for 1) and 2):
1) 1, 3, 6, 10, 15, 21, . . .
2) 1, 3, 7, 15, 31, 63, . . .
Find the third, fourth, and fifth terms for the following recursively defined sequences:
2
3) t1 = 2; t 2 = 4; t n = t n −1 + t n − 2
4) t1 = 5; t 2 = 8; t n = (t n −1 − t n − 2 )
5) Find the first 8 terms of the sequence defined recursively by t1 = 4; t 2 = 8; t n =
t n −1
.
t n−2
6) Use the pattern of terms to predict the value of t1000 for the sequence in problem 5.
Identify the sequences in 7 – 9 as arithmetic, geometric, or neither. Then find an explicit formula for t n
7) 15, 7, −1, −9, . . .
9) 2a − 2b,3a − b,4a,5a + b,...
8) 8, 12, 18, 27, . . .
10) Find t101 for the arithmetic sequence with t 3 = 70 and t 8 = 55 .
11) Find t12 for the geometric sequence with t 3 = 4 and t 5 = 8 .
12) How many terms are in the sequence 18, 24, 30, . . . , 336?
13) Find the value of x if the sequence 2, 8, 3x + 5 is a) arithmetic; b) geometric.
14) Find the value of x if the sequence 4, x,
3x
is a) arithmetic; b) geometric.
2
15) Locate three arithmetic means between −8 and 17.
16) Locate three geometric means between 6 and 54.
17) A ball with 75% recovery is dropped from a height of 12 feet. To the nearest inch, how high does it rise
on the 5th bounce?
18) Two whole numbers m and n have an arithmetic mean of 35 and a geometric mean of 21. Find m − n .
STOP! That's all for today!
Series
19) 4 + 8 + 12 + 16 + 20 + . . .
Express #19 - 20 in Sigma notation.
20) 1 + 8 + 27 + 64 + . . .
For #21 - 28, use an appropriate formula to find each indicated partial sum.
23
21) 16 + 12 + 8 + 4 + . . . + (−56)
22)
(10 − 4m )
∑
m =1
23) S12 for 28 + 34 + 40 + 46 + . . .
24) S 8 for the geometric series with a1 = −2 and
r = 1 .5
∞
25) S 7 for
∑ (1.1t + 8)
t =1
∞
26) S100 for
∑ (1.2)
t
t =1
27) Find the sum of the first 25 terms of an arithmetic series whose third term is 24 and whose common
difference is 3.5.
28) A ball is dropped from a height of 12 ft. Each time it bounces, it rises to a height of 60% of the distance
it fell. Find the total vertical distance that the ball travels by the end of its fifth bounce (that is, just before it
hits the ground for the sixth time.)
29) How many terms in the sequence 2, 4, 6, 8, 10, . . . yield a sum of 132? (Do NOT us a guess-andcheck strategy. Use algebra!)
30) The arithmetic mean of two positive numbers is A, and their geometric mean is G. Find the harmonic
mean of the numbers.