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MATH 1315
EXAM #1 NAME -- ~k:1f'1'~---------------------------
Read the following directions carefully: In all problems, show all work needed and indicate your conclusions in a very clear and organized manner. Unsupported answers may not receive full credit. Please circle the numbers for the four problems that you want me to grade.
1. a) A sequence of numbers is partially described below:
100, 1 1-0^ 1,,"0^ 1 go ,200, roo^ +^?^ d^ .::.-^ z.o^^0
d '" '2..
If this sequence is arithmetic, fill in the blanks with the four missing numbers in the sequence; then find
the 201 st number in the sequence.
b) A sequence of numbers is partially described below:
100, fOO. :;; , 101 , tOI.S , 10'2. , ... ,200, ...
If this sequence is arithmetic and 200 is the 201st number in the sequence, fill in the blanks with the four
missing numbers in the sequence.
\Do + 7..ood =- 2.. de:. t; 0'-'1... ~
c) A sequence of numbers is partially described below:
100, 100 ,400, ---' eoc;> lGooo , ~2-00,
If this sequence is geometric, fill in the blanks with the four missing numbers in the sequence; then find
a formula for the "nth" number in this sequence.
100)C r t l-a-..^ -~-I^ O-O^ -(7..-)--1--',^ I = 2
d) A sequence of numbers is partially described below:
3125, 1.-')00 , 20 0 <> , I!..o , 1 2.-80 , 1024,
If this sequence is geometric, fill in the blanks with the four missing numbers in the sequence.
4 .,
- A club begins with 5 members. Each month, 3 new members join the club and no one drops out. (e.g., by the end of the 2nd month, there are 11 members.)
a) How many members will there be at the end of 50 months?
M~,.,,",, 'L- '\ ~o ~....-~(,d 6 11 (~^ -^7.
v~ ?> (<;0-1) (^) - l-;^ C;^ c;^ M:;;....^ ,~~~
b) How many members will there be at the end of "n" months?
c) Suppose that dues are $1 per month. (New members still pay $1 for the month in which they join, even though they are not members for the whole month.) What is the total amount in dues that the club
will have received during the first 50 months? (e.g., the club receives a total of 8 + il or 19 dollars
dwing the first two months.)
"'-'.^ ~.^ +^ (SS L -:..,^ Lj^ 5'
~ 1'^ ... ,:J."^ S-t>^7
til ,^ l'i^ .tI'
d) Write a general fonnula for the total amount of dues that the club will have received during the first "n" months.
~ ~ I 1 .+ l'f -t 11 -4-. •• +- [~+ ~ (\0'1 -I) )
-')
e) At some point in the future, a rich alumnus of the club donates $5,000 to the club with the following stipulations: 1) The money is to be invested in a savings account that pays 5% per year; and 2) The money is not to be touched for a period of 10 years, after which the money can be used for scholarships. How much money will be available for scholarships after the la-year period?
4. Remember the sequence of triangle figures that we saw on POD #6: A student exploring the same
pattern misinterpreted the problem and saw each of the "upside-dO\vn" triangles as a "hole". In other
words she did not count the triangles that I have marked with an "H" in the figure shown below.
a) Give the number of triangles in the first five figures and a formula for the number of triangles in the
"nth" figure if we count only the unmarked triangles like she did.
Figure number 2 3 4 5
Number of unmarked triangles 10 II)" +'
Figure 1 Fi gure 2 (^) Figure 3
b) Now, using the same sequence of figures shown above (and the student's misinterpretation of the
problem), complete the following table to show the number of "holes" in the first five figures. Then,
create a formula for the number of holes in the "nth" figure.
Figure number (^2 3 4 )
Number of holes o^ (n-t).t"\
t t
c) Now, if we add the unmarked triangles and the "holes", we should generate the pattern we explored in
POD #6. Complete the following table to show the number of triangles (marked or unmarked) in the
first five figures (just as we did on POD #6). Then, create (or re-create) the general rule for determining
the total number of triangles (marked or unmarked) in the "nth" figure.
Figure number 2 3 4 5
Total number of triangles III
d) Now, using algebra, show that the sum ofthe general formulas that you found in a) and b) will result
in the general formula that you found in c).
'L (VI-+-) 1
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1...
. f"
- -
_V_ 1.. t
" -+
h - ...
~l
"1.+,,, +^ 11\ l...
'2.. .... 1. -^ -
'L
- Each function given below is either linear, or exporiential, or quadratic, or "none of these". Indicate which is which and give a short explanation for your answer.
a) (^) x 0 1 2 3 4 5 f(x) (^0) .f .., 3 I .. 8 f , (^15) I q (^24) t (( 35
OUl\d ",,-·h~^ Tlttt.^ el.^ (Clr~c..I-1^ L." IV tt ~^ PA..tte.^ .'^ ,
b) x 0 2 3 4 5 f(x) 8 12 18 27 40.5 60.
)( I. f) 'ilLS- )I. I. S"
\1.- If; ::- I. <; (^) ~ 1. <;;" (^) E)C. p",~e v..-l"'o..l (^) ilL n!.~('s ' ....e.. 1 ::o .....t1~ lY\vel~~ ~ LL. ~t.'<-<---h~ l,"l ~ ~..,to..-J-- l.~.
c) x (^0 2 3 4 ) f(x) 5.42 (^) 4.88 4.34 3.80 3.26 2.
;. S''l -. ~'f -. ~'
Lu,~1V. (^) ~ ~Ir ~,......t. (^) rtltt.L~. 1':1 o...tUl.uL~^ 0-^ ("'.Q'll"-^ ,:..z.^ '^ C-.."h'-h.-+^.
d) x 0 1 2 3 4 5 f(x) 10 12 18 30 50 80 +1- +e;^ t- 11..- + L.J +^ so I- '"l (^) -t- ~ +-& +IC
t"!. 1'6 (^) , f'n.')o. (^) Iv ~ l.}.. (^1) 1"- (^) N O .... l,.. o.c .-fl..'.LSL (^) - p.:;..1;i ~ L' u.I:>,(...
(do <0.'-1 I^ ...^ v-t^ lWo^ ~t..";^^0 ""^ ~^ •^ I,.n'^ h"^ Gl.^ '-<......^ ..t^ 0-0-^1^ ~r
i1AN4-......h L^ ) e} x 0 1 2 3 4 5 f(x) 4 6.5 11.5 19 29 41. t z. .:; (^) s- -I'""1S" + (^10) f- rz.. ~
Q"'\I.-d ",-t-;v 'Du.^ 01.:^ rr^ c;^.^ (L.^ \JL ~'-' ~
x 0 1 2 3 4 5 f(x) 100 108 116.64 125.97 136.05 146. t .o go (^) X t.o~ )(1.0/
lo~
10 0 E)I. eo~'\ 0\ -h.'~ r d/'u"'" -Iv -11M...^ VLffe...kJ^ ~LJ..t^.^ L) "") ',0 '^.
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