arithmetic progression and geometric progression-concept notes

7/30/2019 Arithmetic Progression and Geometric Progression-Concept Notes

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FINPREP - Concept Notes

Chapter Name Arithmetic progression and Geometric Progression

Chapter No. 6

For Private Circulation to registered students. Page 1 of 11

FINPREP

A CPT preparatory program from

Concept Notes

Subject: Quantitative Aptitude


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Chapter No. 6


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Chapter No. 6


Arithmetic progression and Geometric Progression

SEQUENCES AND SERIES

LEARNING OBJECTIVES:

To distinguish between sequence and series. To introduce different types of sequences and series. To define and develop the properties of Arithmetic Progression (AP). To define and develop the properties of Geometric Progression (GP). To determine the sum, the sum of the squares and the sum of the cubes of natural numbers. These concepts are used in many real life problems like simple interest, compound interest,

annuities, population growth, recurring deposits etc.

SEQUENCE:

An ordered collection of numbers a1, a2, a3, a4, .., an, . is a sequence if according to some

definite rule or law, there is a definite value of an, called the term or element of the sequence,

corresponding to any value of the natural no.

A finite sequence a1, a2, a3, ., an is denoted by

1

n

iia

and an infinite sequence a1, a2,

a3,an is denoted by 1nn

a

or simply by {an} where an is the nth element of the sequence.

Examples of infinite sequence:

1) The sequence {1/n} is 1, , 1/3, , 2) The sequence {( - 1) nn } is 1, 2, - 3, 4, - 5,

Examples of finite sequence:

1) A sequence of even positive integers within 12 i.e., 2, 4, 6, 102) A sequence of odd positive integers within 11 i.e., 1, 3, 5, 7, 9 etc.

SERIES:

An expression of the form a1 + a2+ . + an+ .. Which is the sum of the elements of thesequence {an} is called a series. If the series contains a finite number of elements, it is called afinite

series, otherwise called an infinite series.


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Chapter No. 6


ARITHMETIC PROGRESSION (AP):

A sequence a1, a2, a3, an is called an Arithmetic Progression (AP) when a2 a1 = a3 a2= .. =

an an 1. That means A.P is a sequence in which each term is obtained by adding a constant d to thepreceding term. This constant d is called the common difference of the A.P. If 3 numbers a, b, c is in

A.P., we say

b a = c b or a + c = 2b; b is called the arithmetic mean between a and c.

Now in general an A.P series can be written as

a, a + d, a + 2d, a + 3d, ..

When a is the 1st

term and d is the common difference.

nth term (tn) = a + (n 1) d, where n is the position no. of the term..

Sum of the first n terms:

Let S be the Sum, a be the 1st

term and l the last term of an A.P. If the number of terms are n,

then tn = l. Let d be the common difference of the A.P.

Now S = a + ( a + d) + (a + 2d) +..+ (l 2d) + (l d) + l

Again S = l + (l d) + (l2d) + + (a + 2d) + (a + d) + a

Sum of 1st

n natural numbers:

S = 1 + 2 + 3 + .. + (n 2) + (n 1) + n

Sum of 1st

n odd number is S = 1 + 3 + 5 + + (2n 1)

Sum of 1st

n odd numbers is S = n2

Sum of the squares of the 1st

n natural numbers is S = n (n = 1) (2n + 1) / 6

Sum of the cubes of 1st

n natural number is S =

2( 1)

2

n n

Note:

i) AnAP remains anAP if a constant quantity is added to or subtracted from each term of theAP.ii) It also remains anAP if each term of theAP is multiplied or divided by a constant quantity.iii) If 3 terms are inAP, they are of the form ad, a, a + div) If 4 terms are inAP, they are of the form a 3d, ad, a, a + d, a + 3dv) Ift1, t2,..are inAP with CDdthen t1, t4, t7,. are inAP with CD 3d.


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Chapter No. 6


vi) Ift1, t2, t3,,tn are inAP then t1 + tn = t2 + tn 1 = t3 + tn 2. Ifn is odd say n = 2m + 1 themiddle term is tm + 1 and 2tm + 1 = t1 + tn.

vii) If three terms are in GP, they are of the form ar

, a, ar

viii) In anAP,SnSn 1 = Tn.ix) Ifa1, a2, .., an are inAP, then

1

1

a,

2

1

a, .,

1

na

are said to be HP (harmonic progression)

Arithmetic Mean (AM):

Xis said to be theAM between a and b ifa,xand b are inAP.

xa = b x

(i.e.) 2x= a + b

x=( )

2

a b

Arithmetic Means:

x1, x2, x3, , xn are called n arithmetic means between a and b ifa, x1, x2, ,xn, b are inAP.

There are (n + 2) terms in thisAP.

Thenx1 + x2+ ..+ xn = n2

a b

i.e. Sum of n A.Ms is nothing but n times single AM between a and b.

Geometric Progression:

If in a sequence of terms each term is constant multiple of the proceeding term, then the

sequence is called a Geometric Progression (G.P.). The constant multiplier is called thecommon ratio. A

series, in which the ratio of any term to its preceding term is constant, is called a Geometric Progression

(GP). The constant quantity is called the common ratio (r).

Examples:

1. 2, 4, 8, 16, ..2. 4, 12, 36, 108, .3. In 5, 15, 45, 135, .. common ratio is 15/5 = 34. In 1, , , 1/8,. common ratio is (1/2) / 1 =


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Chapter No. 6


Formula for the nth

term

Let a be the first term and rbe the common ratio of a GP.

Then the series is a, ar, ar2,

1st term = T1 = a

2nd

term = T2 = ar

3rd term = T3 = ar2

4th

term = T4 = ar3

Proceeding in this way we get the nth term as Tn = arn 1

FORMULA FOT THE SUM TO n TERMS

Let a be the first term and rbe the common ratio of a GP.

Let Sn denote the sum to n terms.

Then,

Sn= a + ar+ ar2+. + arn

1(1)

Multiplying both sides by r,

rSn= ar+ ar2

+ ar3+. + ar

n 1+ ar

n(2)

Subtracting (2) from (1),

Sn(1 r) = aarn

= a( 1rn)

Sn = (1 )1

na r

r

(3)

=( 1)

1

na r

r

(4)

Formula (3) is used if the common ratio is less than 1 and formula (4) is used if the common ratio is

greater than one.


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Chapter No. 6


SUM TO INFINITY

If |r| < 1, as n, rn 0.

S =1

a

r

If |r| >1, S does not exist.

Geometric Mean (GM):

xis said to be the GM between a and b ifa, x, b are in GP.

x

a =

b

x (ie)x2

= ab

n GEOMETRIC MEANS BETWEEN a AND b

x1, x2, x3, ., xn are said to be the GMs between a and b ifa, x1, x2, ..,xn, b are in GP. There

are (n + 2) terms in this GP. Let rbe the common ratio. Thenx1 .x2.xn = n

ab = (ab)n/2

i.e. product of n GMs between a and b is nth power of single GM.

SOLVED EXAMPLES

Example 1:

Find the 6thterm of the sequence 1, 4, 9, 16, 25, ..

Solution

The terms are 12, 2

2, 3

2,..

the 6th term of the sequence = 62 = 36.

Example 2:

If the nth

term of a sequence is n2

+ 2n + 2 find the 6th

term.

Solution

Given that the nth term is

Tn = n2

+ 2n + 2

Put n = 6,

T6 = 62 + 12 + 2 = 36 + 12 + 2 = 50.


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Chapter No. 6


Example 3:

Find the 7th

term of the series 15, 12, 9, .

Solution

The given series is anAP.

Tn = a + (n 1)d

T7= 15 + (7 1)( - 3)

= 15 - 18 = - 3

Example 4:

Which term ofthe series 7, 5, 3, 1, is 37?

Solution

Tn = a + (n 1)d

- 37 = 7 + (n 1) ( - 2)

= 7 2n + 2

2n = 46 n = 23.

Example 5:

The rate of monthly salary of a person increases annually inAP. It is known that he was drawing

Rs 200 and Rs 380 during 11th

year and 29th

year respectively.

Solutiona + 10d= 200

a + 28d= 380

18d = 180 or d = 10

Hence a + 100 = 200

a = 100

Starting salary = Rs 100.

Example 6:

The sum of three numbers inAP is 12. Find the middle number.

Solution

Let the three numbers be a d, a, a + d

a d + a + a + d = 12

3a = 12

a = 4

Middle number is 4.


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Chapter No. 6


Example 7:

Find the sum of the first three terms of the series given Tn= n22n.

Solution

Tn= n22n

T1 = - 1

T2 = 4 4 = 0

T3 = 9 6 = 3

T1 + T2 + T3 = - 1 + 0 + 3 = 2.

Example 8:

If 8x+ 4, 6x 2 and 2x+ 7 are inAP findx.

SolutionIfT1, T2, T3 are inAP then

2T2 = T1 + T3

2(6x 2) = 8x+ 4 + 2x+ 7

12x 4 = 10x+ 11

2x= 15

x= 15/2.

Example 9:

The sum of four numbers are inAP is 16. Find the sum of the two middle numbers.

Solution

Let the numbers be a 3d, ad, a + d, a + 3d.

a 3d+ ad+ a + d+ a + 3d= 16

4a = 16

a = 4

Sum of the middle terms is

a d+ a + d= 2a = 8.

Example 10:

Ifa, b, c are inAP then show that b + c a, c + a b, a + b c are inAP.

Solution

a, b, c are inAP.

-a, - b, - c are inAP.

- 2a, - 2b, - 2c are inAP.

Add a + b + c to each term

b + ca, c + ab, a + bc are inAP.


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Chapter No. 6


Example 11:

If b ca , c a

b , a b

c are inAP then show that 1/a, 1/b, 1/c are inAP.

Solution

b c

a

,

c a

b

,

a b

c

are inAP

Add 1 to each term.

a b c

a

,

a b c

b

,

a b c

c

are inAP.

Divide each term by a + b + c.

Then 1/a, 1/b, 1/c are inAP.

Example 12:

If log a, log b, log c are inAP show that a, b, c are in GP.

Solution

Log a, log b, log c are inAP.

2 log b = log a + log c

b2

= ac

Hence a, b, c are in GP.

Example 13:

Ift1, t2,t3, are inAP then t1, t5, t9 are also inAP.

Solution

a, a + d, a + 2d, are inAP.

Then the sequence t1, t5, t9,..area, a + 4d, a + 8d,

This is also inAP with common difference 4d.

Example 14:

If thepth

term of anAP is q and qth

term isp; the rth

term of theAP is

(a)p q + r (b)p + q r (c)p q r (d)p + q r + 1


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Chapter No. 6


Solution

a + (p 1) d= q(1)

a + (q 1) d=p ..(2)Subtracting (pq) d= qp or d= - 1.

From (1), a = q +p 1

Tr =p + q 1 + (r 1) ( - 1)

=p + q 1r+ 1 =p + qr.

Example 15:

Thepth

term of anAP is q and qth

term isp. The (p + q)th

term is

(a)p + q (b)pq (c)p q (d) 0Solution

a + (p 1) d= q

a + (q 1) d=p

Subtracting (pq) d= qp (or) d= -1

a = q +p 1

Tp + q = a + (p + q 1) d

= (p + q 1) (p + q 1)

= 0.

arithmetic progression and geometric progression-concept notes

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