Teooher Lesson
uNrr 1 - ARrrHMErc & cEoMErRIc ffillr*3L?
NDle 5
MA40: ALGEBRA 2
Name:
Period:
Task #4 - Counting Dots (Finding Rules for nth Term of a Sequence)
Common Core: HS.F-1F.3, HS.F-BF .1a, 2, HS.F-LE.2
DERIVE THE EXPLICIT FORMULA FOR THE nth TERM OF AN ARITHMETIC SEQUENCE
Finding the formula for any term of an arithmetic sequence is essentially looking for a pattern and
making a generalization that fits any and every term in the sequence. Complete the table using
the drawing the dot problem. Find the explicit formula that describes the n'u term.
1.
a
At one
minute
At the
Reoinnino
fiiffi
'%J
H
:.:
tt%.J
..ttt%
at'\
Attwo
At three
At four
minutes
minutes
minrrtes
# $ffi
At the
1t'
beginning
term
At one
2^o
minute
term
At two
3td
minutes
term
At three
minutes
term
4'n
At four
5tn
minutes
term
,
L
=l + (o.r)
5
=7+1
=lr(r.q)
9
= I + tl +t{
= l+tl +rl +{
= I + (a,t)
r3
t1
= l+rl+rl+e{
= l+ (3.,f)
I +(q .,)
+Ll
=
t$ *Ym ?(
At fourteen
15'n
min.
term
57
= I +(rq.q)
I
At
n-1
minutes
2.
Ar
*F
\
l+[n-),r]
+ tln-{
On = {n -3
On
itr rh-D'd)
n'u
term
use this exolicit
the nun
number
explicit formula to
Use algebra to clean up this formula (simplify)
--of dots at 40 minutes. Use this explicit formula to find the number of dots at 67 minutes.
0.,,=
O'
n- I\.*rrncr
t0rnrntics
It = tll
=lHt
=) {l$trt",
67rrrn,;{€
s =} Ggi/!. tcrmr n= 68
oes=q(6s)
= lbl
-3 = e61
iS 16l
to represent the common difference, rewrite the
equation derived in problem #2by substituting in a, and d. Tl,i
ntb ter""
Rule for Finding the nth Term of an Arithmetic Sequence.
ExDlirit Rui"
loccr,tion in Se2r.lencc
61n=
o$ Ar1$lrnetic SeVoancl
- nQ ter"n ,[-
- $
+orvn
AL
qO rurn,rlce t{ncce
q6rwon
dtffucncc
lr(n-r).{
o,lffi':,.
0,,#*
6;
Qn= 0,
f (n -t) d
F$R THE nth TERM OF AN ARITHMETIC SEQUENCE
Complete the table using the drawingntne dot problem. Find the recursive formula that
describes the nh term:
DERTVE THE RECURSIVE FORMULA
4.
ffiu$**"'
rpos'
6
qp,tX-
At
-*uJ *0"-
Minutes
::.i
:.:
At
the
Beginmng
attt\
...'
At three
At four
minute
minutes
minutes
minutes
minutes
A four
minutes
At fourteen
minutes
4
5
15
t3
t1
At two
At three
Beginning
minute
minutes
1
2
3
5
L
at\
%
At two
At one
# of Dots
t
one
At the
Term
tJ
9
Al
n-
1 minutes
n'h
tl
Gn =O"-, *
57
Compare the table in problems #1 and til. \trthy is the table in problem #1 better suited to find
an explicit formula? Why is the table in problem #4 better suited to find a recursive formula?
uJt'
tic-deqi:e"cc'
its
luatfio5
to
GCcoratrng
on
tecrn
c*
An erpricis famura. descrrbes
bc**recn the eo'i-ond'fCc'taocc
Lr,c fu*s a6 o. Eurn (in tabra *,J "1ir,
SC pwvloos tcr"' G)'
On blceol
,1 orraL {rrc porrlb... n. A rgcurg,vi {nrnuto d"sliri"t'o
'''..
ro, aidcLn1obtc*{) qllews os lo gcc this nclah'a'r5rrrp'
brvr'i-6r.*d0+6'id.
Using a, to r6pieseit the first term and d to represent the common difference, rewrite the
5.
--
fi;;i;;'n-Tn
; ;; P;;;i""."J"t'1i^onrp
t.t-
I.;^;
equation derived in problem #4by substituting in a, and d.
Rule for Finding the nth Term of an Arithmetic Sequence.
Or =
h- locaHon rn
19t 66rm a{
On- n+I ter"*a ot-Carnrrno,l On a Qn-r +
a?vev,c<
- di{[crc.c<
tcc-r
A, - l9l
-,1",il,""",J,"ii;;,
,
I <- T,[i'#,"^y'
I
I
Recr"srvc Rtle ftr, n&terr^
9$ fleilhmcttc Sqrocncg
A.t = l!3 tcrr'.'
Pg:1l ,' ',j n*:,?'T*
Ar'r=Orr-rtd
r,'ffif#Hi:'It
6minu*h*-r#d
mi?n"
nltrutcs =) l6ib {,crwr
"
find the number of dots at l$Ufiinutes.
167ffi]
bmrnutct tE
f*
Fr'not qrr l---h= lb
.f 0z 300+t{
O"g
15
Q,o=As
+
thoea
=
Ors = 57
At tu mrntrt€5
:|l+H
ddj
11
6rnrrruf6e
=7 Tbh.l,r
FrnoL o.7
as
= 17 +Ll
fitrOtt i9
Qbo 4,s t Y
= 57 rY-- 6t
Qt dols
=a5
=al
8. Find the explicit and recursive rules for the nth term of the given sequence. Ju
work. Expttort RulC
gg, 93,87, 81, 75, ... Ar-=O, tCn-f)d
Or= 91 On e nt{rc,n On = 9i r (n-r) 0C)
d=-b n-lacah'onrnrr1. _ =qqtC6r)+
ta61fr6ecocrrve Ru\
by showing your
9.
Find the explicit and
common difference d.
Exph'cit Rule
An*
0.15n
t o.es
0n = -6nt
105
Q, =9j
0.r, = Qn-l+ C6)
t.\rv\c
Exph'orl Rtl\t
On: O, * Cn-r)d \
0n a 0,5 + (n-r) [o.rs)
<0,5
+ o.lsn
-0,15
with the first term
RecurtrvoRolc
O.: l!!tcct"'
Or.raflrrn+ot
q and the
L
e
= 0,5
On*On{ + p.es
Qr
DERIVE THE EXPLICIT FORMULA FOR THE Nth TERM OF A GEOMETRIC SEQUENCE
10. Finding the formula for any term of a geometric sequence is essentially looking for a pattern and
making a generalization that fits any and every term in the sequence. Complete the table using
the drawing the dot problem. Find the explicit formula that describes the r'tu term.
aa
At the beginning
eo
aa aa
ao
ao
aa
ao
ao aa
At one minute
Attwo mindes
taoa
aaaa
aaaa
oaaa
aaaa
aaaa
aoaa
aoaa
atoa
aaaa
aaoa aooo
oaaa
aaoa
aooo
aaaa
aaaa aaaa
At three minutes
At four minutes
.l
At the
beginning
term
3
At one
minute
znd
term
(.
At two
minutes
term
At three
minutes
term
At four
5th
minutes
term
1"t
a'l
l
=32)e
= 3,e'
=J,e^
2 3'l'
q8
= 3.1.)
=
],r*
=
3,2*
la
3'd
4th
? 3,)o
=3
= 3'a
=3I
) '^
lg trvncs
At fourteen
1sth
min.
term
lIt5a
/n-r
Al
n-l
s
e 3.atn-o \
n'o
term
minutes
11.
number of dots at20 minutes. Use
minutes.
lomrnufus =]
nr
G.na
) *r"*l
?.r'
ltlt
o,, 111'1,_,
=
3,
ltls,1
this#
?t
&utatotindM
formula to find the number
15 mrnulc' =) 2o*ctr'"
lO mrrrutr5
|ucca is
J, tu9,71.3
dols
lb
ato= 3'ft-t'
n=
= lMtb&,Tqb
q
At
15 mr"ticS
ttre,rg is
loot|bS,,qb
dofg
rewrite the
to represent the first term andfa to represent the common
-t.
This new e(uation is the
equation derived in problem #2by substituting in q and
€xph'cr*
Rule for Finding the nth Term of an Geometric
12. Using
- loccoti on o.sfli,
On - n+) tcrrryr
O, - tlj tcrr-. rutro
4f- clnvnon ffimC
h
ffi*,
Gn
= 3 . e"'t)
,$ \-'
Exptic,rf Rvle
6" nr!tct*
oi Geo,n"*,"r1:3ucb(c
An= o,' /n-t
nffg
DERTVE THE RECURSIVE FORMULA FOR THE nth TERM OF A GEOII,IETHC SEQUENCE
13. Complete the table using the drawing the dot problem. Find the recursive formula that describes
lhe nh term.
aa
At thebeginning
Minutes
Term
# of Dots
oa
aa ao
aa
aa
aa
aa aa
aa
At
Attwo minliles
one minute
aaaa
aaaa
aoaa
aaaa
aaaa' aaao
At three minuGs
At fourminutes
At the
At one
At two
At three
minute
minutes
minutes
A four
minutes
At fourteen
Beginning
1
2
3
4
5
15
6
ra
)q
3
aaao
aaaa
aaaa
aoaa
aoaa
aaaa
oaoo
ooaa
oaaa
aoaa
aaaa aaaa
r.l
Al
minutes
=On{.I
formula.
,t'ru*roJJ *I,a| #- tc*n
14. Describe how you used the table to find the recursive
ffi;fir?"+;;;;;Ir
81 cannporing
derrved. by iu*;prying llrc Pnc'vious
1*'^
by
1 minutes
n'h
{ 9,t51
8
n-
2'
0(rtv(
\De$r'eXec wtrrc
,r\A%qff
,c
CfFt
I i'--'.lr+ 'f
I lce this rearrsive formrrla to fina fhe nrrmber of dots at f %inutes t.lse this l.."rr",rJ*ril3#
15.Usethisrecursiveformulatofu.tdthenumberofdotsatIminutes.UsethisrecurSlvW
findthen,moeiotJ;i;;i.tfiinutes.]5-.".:I1:'a+W
4t 6mrrulcl
-l Qrs*
'r
Ttrrn
19,151
t-'n*=s
i5
l
thecc
tv,oo.
=
R.currrvg R,rle
[ffi*',lo
""^
dots
1r
o''
0s q s 1;146,
ji":":',
16.
@)
I
=
l. r,-,il',li
r=
W
Find the explicit and recursive rules for the nth term of the given sequence. Justi-fy your answer
by showing your work. EXplicrt Rule
ilecurotve Rulc
Q,= 1!r tcrw'
-3, 6, -12, 24,
On=O,. I.(tq)
lffi
...
0'=-3
]=tl:]f,
-]
On
= O*r'
f
cn= o*r'(-1)
I
4n = n$ tcc't'
[ = lacotrox
17.
Find the explicit and recursive rules for the nth term of the sequence with the first term
the common ratio
Recucsrva Rute
r.
a,=3f -t'2
I
EXpllcr$ Rulg
4q=0,'
Qr, =
(l)
ftn-[
'(t
4. = lsJf $crw.
Ar,zQr..t'F
q
and
O,= 3
O.= onr'(+)