Progression

Arithmetic and Geometric Progression

1. UNIT 3
Progressions
Mr.T.SOMASUNDARAM
ASSISTANT PROFESSOR
DEPARTMENT OF MANAGEMENT
KRISTU JAYANTI COLLEGE,
BANGALORE

2. UNIT 3
PROGRESSIONS
Introduction,ArithmeticProgression,finding
then
th
termofAPandsumton
th
termofAP,
Insertionofarithmeticmeansingiventermsof
APandrepresentationofAP,Geometric
Progression,findingn
th
termofGP,sumton
th
termofGP,Insertionofgeometricmeansin
givengeometricprogressionandalso
representationofGP.

3. INTRODUCTION
SequenceRealNumbers:
“Asetofrealnumberswrittenin
successionaccordingtosomeruleissaid
toformasequenceofrealnumbers.The
successivenumbersinthesequenceare
calleditstermsorelements.”
Sequence{X
n}=X
1,X
2,X
3,……..,
X
n,…………… .
Thenumber’sX
1,X
2,X
3,arecalled
elementsofsequence{X
n}andX
nis
callednthelementofsequence{X
n}.

4. Finite Sequence:
“The no. of terms is finite then such
a sequence is called a finite sequence.”
(i.e.) finite sequence has a last element.
(E.g.) {X
n}= 2, 4, 6, 8, ……….32.
Infinite Sequence:
“Infinite sequence is sequence which has
no last term and it is denoted by ∑ X
n.
(E.g.) X
1+ X
2+ X
3+……..+X
n
+…………… ..

5. Arithmetic Progression
ArithmeticProgression:
ArithmeticProgressionisasequenceofrealnumbersin
whicheachelementofthesequenceisobtainedbyadding(or
subtracting)thesamenumber‘d’toitspreviouselement.
Definition:
“Asequenceofnumbersinwhichdifferentelements
(exceptfirstelement)arewrittenbyincreasingordecreasing
itspreviouselementsbythesamequantityiscalledan
ArithmeticProgression(A.P.)”
A.P.=a,a+d,a+2d,a+3d,……………
where‘a’iscalledfirstelement,‘d’isfixedno.bywhich
elementareincreasedordecreasedanditisalsocalledas
differences.

6. Arithmetic Progression
(i.e.)1
st
element=a
2
nd
element=a+d
3
rd
element=a+2d
…………………… .
10
th
element=a+9d
n
th
elementofA.P.=a+(n–1)dandit
isdenotedbyT
n.
Formula:
T
n=a+(n-1)d

7. Arithmetic Progression
Sum to n terms of an A.P.:
Formula:
Sn = n / 2 [2a + (n –1) d]
(Or)
Sn = n / 2 [a + l]
where l = a + (n –1) d

8. Arithmetic Progression
Exercise problems:
1. Find the 20
th
term of A.P. 2, 6, 10, ……….
2. Which term of A.P. is 5, 13, 21, …… is 181?
3. Find the common difference of A.P. where first
term is 5 and 11
th
term is 25.
Classwork problems:
1. Find the 6
th
, 8
th
and 17
th
term of A.P. whose nth
term is 4n –3.
2. If the first term of A.P. is 2, the 20
th
term is 59, find
32
nd
term?
3. If 7 times 7
th
term of A.P. is equal to 11 times its
11
th
term show that 18
th
term of A.P.is zero.

9. Arithmetic Progression

10. Arithmetic Progression
Homework problems:
1. Find the 25
th
term of A.P. 0.3, 1, 1.7, ……….
2. The 10
th
term of A.P. is 2 and 16
th
term is –10. Find
the 11
th
term.
3. Determine the 2
nd
term and the r
th
term of A.P.,
whose 6
th
term is 12 and 8
th
term is 22.
4. Find the three numbers which are in A.P. whose
sum is 12 and the sum of their cubes is 408.

11. Sum to n
th
terms of an A.P.

12. Sum to n
th
terms of an A.P.

13. Sum to n
th
terms of an A.P.
Homeworkproblems:
1.FindthesumofthefollowingA.P.2,6,10,……to
50
th
term.
2.ThesumofnelementsofA.P.25,22,19,16,……
is116.Findthenumberoftermsandthelastterm.
3.Findthesumofthefirsthundredevennatural
numbersdivisibleby5.

14. Application Problems (A.P.)
Exerciseproblems:
1.ApersonbuyseveryyearBank’scashcertificateof
valueexceedingthelastyear’spurchaseby250.
After20years,hefindsthatthetotalvalueofthe
certificatespurchasedbyhimisRs.72,500.Find
thevalueofthecertificatepurchasedbyhima)in
thefirstyearandb)inthe13
th
year.
2.Amanufacturerofradiosetsproduced600unitsin
the3
rd
yearand700unitsinthe7
th
year.Assuming
theproductionuniformlyincreasedbyafixed
numbereveryyear,findi)productioninfirstyear,
ii)totalproductionin7yearsandiii)productionin
10
th
year.

15. Application Problems (A.P.)
Homeworkproblems:
1.Acontractoragreestosinkawell250feetdeepata
costofRs.2.70for1
st
foot,Rs.2.85for2
nd
footand
anextra15paiseforadditionalfoot.Findthecost
ofthelastfootandthetotalcost.
2.AclubconsistsofmemberswhoseagesinA.P.,the
commondifferencebeing3months.Iftheyoungest
memberoftheclubisjust7yearsoldandthesum
oftheagesofallmembersis250years,findthe
numberofmembersintheclub.

16. Geometric Progression
GeometricProgression:
Definition:
“AGeometricProgressionisasequenceofnumbers
inwhichtheratioofeveryelementtoitspreviouselement
isafixedconstant.Thisfixedconstantiscalledcommon
ratio.”
G.P.elementsarea,ar,ar
2
,ar
3
,…ar
n-1
,…
where‘a’isthefirstelementandristhecommonratioofA.P.
Tofindcommonratio,
r=ar/r(or)r=ar
2
/ar(or)ar
3
/ar
2
=
…….=ar
n-1
/ar
n-2
……..=r
wherear
n-1
iscalledthenthelementorgeneraltermsand
itisdenotedbyT
n.

17. Geometric Progression
Sum to n terms of an A.P.:
Formula:
S
n= a (1 –r
n
) / (1 -r)
where ‘a’ is first element and ‘r’is the
common ratio.

18. Geometric Progression (G.P.)

19. Geometric Progression (G.P.)

20. Geometric Progression (G.P.)
Exercise problems:
1. The ratio of 9
th
element of a G.P. to the 6
th
element
is –8 and the 5
th
element is 16. Find the G.P.
2. The 10
th
element of a G.P. is double the 12
th
element. If the 3
rd
element is 6, find the 5
th
element.
3. Which element of G.P. 1, -2, 4, -8, …….. Is 1024.
4. The three numbers whose sum is 18 are in A.P. If 2,
4 and 11 are added to them respectively, the
resulting numbers are in G.P. Find the numbers.

21. Geometric Progression (G.P.)
Classwork problems:
1. The second, third and sixth elements of an A.P. are
consecutive terms of a G.P. Find the common ratio
of the G.P.?
2. Find the three numbers which are in G.P., if their
sum is 28 and their product is 512.

22. Geometric Progression (G.P.)

23. Sum to n
th
terms of an G.P.

24. Sum to n
th
terms of an G.P.
Classworkproblems:
1.ThesumoffirsttenelementsofaG.P.isequalto
244timesthesumoffiveelements.Findthe
commonratio.
2.ThefirstandlastelementsofaG.P.are3and768
respectivelyandthesumis1533.Findthecommon
ratioandthenumberofterms.
3.Findthesumtontermsof7+77+777+7777+
…..

25. Sum to n
th
terms of an G.P.
Homeworkproblems:
1.FindthesumoftheG.P.27,9,3,1,….to8
element.
2.HowmanyelementsoftheG.P.1,2,4,8,………
mustbetakenamountto255?
3.Findthesumtontermsof1+11+111+1111+
…...

26. Insertion of AM and GM

27. Arithmetic & Geometric means

28. End of the Unit 3
Thank You