Mathematical Investigations IV
Name
Mathematical Investigations IV
Iteration Forever
LOOKING BACK
Show proper use of notation, formulas, and techniques you have learned in this unit.
1.
Find the 180th term and the sum of the first 180 terms in the arithmetic series: 2 + 6 + 10 . . .
2.
a. Insert two geometric means between: 27 and – 125.
b. Find the 12th term of the sequence determined in (a) and then find the sum, S12 .
c. Find the sum of the infinite series determined by the terms of the sequence in (a).
3.
Find the 24th term of the harmonic sequence: 12, 6, 4, 3, . . .
4.
Find the next three terms of the sequence: 1, 2,
5.
Write each of the following repeating decimals as a reduced common fraction.
a. 0.723723723. . .
b. 0.724242424. . .
Seq & Ser. 10.1
7,
10,
13, 4,
.
Rev. S08
Mathematical Investigations IV
Name
6.
A crazy fly flies from the ground, up 18 feet, then down 12 feet, then up 8 feet, and so on,
forever, traveling 2/3 of the previous distance each time.
a. How far does the fly travel
b. What height above the ground is the fly
all together?
approaching?
7.
Find the sum of each series.

a.


1k
k0
3  2k
b.

d.
2
49
c.
8.
j
2 1
j
j 1 5
10
 5 j  2 
j 21
k 1
Insert four arithmetic means between
the bases, 18 and 27, to determine the
lengths of the four equally spaced
parallels in the trapezoid.
k
 k2

18
27
Seq & Ser. 10.2
Rev. S08
Mathematical Investigations IV
Name
9.
Write each of the following in sigma notation, and then evaluate the sums if possible.
a. 11(12)(13) + 12(13)(14) + . . . + 30(31)(32)
b.
10.
11.
3 2 4 8
  
 . ..
5 5 15 45
c.
Find the following sums:
20 
1
1 

a.  

2n  1 2n  3
n1
Solve for x:
1–1+1–1+...
b.
4
4
j 1
k 1
log 2
1
2
3
127
 log 2  log 2  . . .  log 2
2
3
4
128
 (2  jx)   (4k  x)
Seq & Ser. 10.3
Rev. S08
Mathematical Investigations IV
Name
12.
A
Given:
B1 is the midpoint of BC.
B2 is the midpoint of B1C.
B3 is the midpoint of B2C
… and so on until …
Bn is the midpoint of Bn-1C.
17
B
Find the sum of the lengths of all the
vertical segments.
13.
B1
B2
B3
BC = 15
 c

Consider the sequence xn =  2xn1
 5  12
a. Find the fixed point.
if n  1
if n  1
b. Is this a convergent fixed point? Justify.
c. Is the value of c important?
14.
Let An equal the area of the shaded region for value n.
 
a. Find the sequence An
6
n1
n+1
n
100
b. Find the sum
A
n
n1
Seq & Ser. 10.4
Rev. S08
C
Mathematical Investigations IV
Name
15. Given the sequence of figures:
a.
Draw the next two figures in the sequence.
b.
Find a sequence whose terms equal the number of dots in each figure.
c.
Derive a formula for the nth term of this sequence of numbers.
16. Dough Boy puts $5.00 in the bank at the end of week one, $10.00 at the end of week two,
$15.00 at the end of week three, etc. If this pattern continues,
a. how much dough does Dough Boy put in the bank at the end of week 200?
b.
17.
how much dough has Dough Boy banked all together by the end of week 200?
Rewrite 22  42  62  82  102 
 602 with -notation and then compute the sum.
Seq & Ser. 10.5
Rev. S08
Mathematical Investigations IV
Name
k
if n  1

18. Consider the sequence an  
2an1  3 mod 7 if n  1
a. Complete the table below to find terms based on different seeds.
a1 = k
a2
a3
a4
a5
a6
a7
0
1
2
3
4
5
6
b. Complete the "clock" diagram started to the
right to show the orbits.
0
1
6
2
5
4
Seq & Ser. 10.6
3
Rev. S08