ML Aggarwal Solution Class 10 Chapter 9 Arithmetic and Geometric Progressions Exercise 9.5

 Exercise 9.5

Question 1

Find the sum of:

(i) 20 terms of the series 2 + 6 + 18 + …

(ii) 10 terms of series 1 + √3 + 3 + …

(iii) 6 terms of the GP. 1,23,49,

(iv) 20 terms of the GP. 0.15, 0.015, 0.0015,…

(v) 100 terms of the series 0.7 + 0.07 + 0.007 +…

(vi) 5 terms and n terms of the series 1+23+49+

(vii) n terms of the G.P. √7, √21, 3√7, …

(viii)n terms of the G.P. 1, – a, a², – a³, … (a ≠ – 1)

(ix) n terms of the G.P. x3,x5,x7, … (x ≠ ±1).

Sol :

(i) 2 + 6 + 18 + … 20 terms

Here, a = 2, r = 3, n = 20, r > 1

S20=a(rn1)r1

=2(3201)31=2(3201)2=3201


(ii) 1+3+3+10 terms

Here, a=1,r=3,n=10,r>1

S10=a(rn1)r1=1[(3)101]31

=(3101)(3+1)(31)(3+1)

(Rationalising the denominator)

=(351)(3+1)31=(2431)(3+1)2

=242(3+1)2=121(3+1)


(iii) 1,23,49,6 terms

Here, a=1,r=23,n=6,r<1

S6=a(1rn)1r=1[1(23)6]1+23

=35(1(2)636)

=35[164729]=35[72964729]

=35×665729=133243


(iv) 0.15,0.015,0.0015, ... upto 20 terms

Here, a=0.15,r=110,n=20

S20=a(1rn)1r (r<1)

=0.15[1(110)20]1110

=15×10100×9[111020]

=530[111020]=16[111020]


(v) 0.7+0.07+0.007+... 100 terms

Here, a=0.7,r=110,n=100,r<1

S100=a(1rn)1r

=0.7[1(110)100]1110

=7×1010×9(1110100)=79(1110100)


(vi) 1+23+49+n terms, 5 terms

Here, a=1,r=23÷1=23,n=5,n,(r<1)

Sn=a(1rn)1r=1[1(23)n]123

=1×31[1(23)n]=3[1(23)n]

and S5=3[1(23)5]=3[132243]

=3[24332243]=21181


(vii) 7,21,3¯7,

Here, a=7,r=217=7×37=3

r>1,n=n terms

Sn=a(rn1)r1Sn=7[(3)n1]31

Rationalising the denominator.

Sn=7[(3)n1]31×3+13+1

=7[(3)n1](3+1)(3)2(1)2

=7[(3)n1](3+1)31

=72[(3)n1](3+1)


(viii) 1,a,a2,a3,(a1) upto n terms

Here, a=1, r=-a

Sn=a(1rn)1r=1[1(a)n]1(a)=1(a)n1+a


(ix) x3,x5,x7,(x±1)

Here, a=x3,r=x2

Sn=a(1rn)1r=x3[1(x2)n]1x2 if r<1

=x3(1x2n)1x2


or Sn=a(rn1)1r=x3[(x2)n1]x21

=x3(x2n1)x21


Question 2

Find the sum of the first 10 terms of the geometric series

√2 + √6 + √18 + ….

Sol :

√2 + √6 + √18 + ….

Here, a = √2 , r = √3, r > 1

S10=a(rn1)r1
=2[(3)101]31=231[(3)51]

=231(2431)=231×242

=2(3+1)×242(31)(3+1)

(Rationalising the denominator)

=242(6+2)31=121(6+2)2

=121(6+2)


Question 3

Find the sum of the series 8127+9127

Sol :

8127+9127

Here, a=81,r=2781=13,l=127,r<1

Sn=alr1r

=81+127×131+13

=8118143

=6561181×43

=6560×381×4=164027


Question 4

The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then find its first term.

Sol :

In a G.P.

Tn=128
 Sn=255
r=2
Let a be the first term, then
Tn=arn1
128=a2n1
a=1282n1..(i)

Sn=a(rn1)r1

255=a(2n1)21

255=a(2n1)

a=2552n1..(ii)

From (i) and (ii)

2552n1=1282n1

255×2n1=128(2n1)

255×2n1=128×2n128

255×2n2=128×2n128

255×2n=256×2n256

256×2n255×2n=256

2n=256=28

Comparing, we get

n=8

Now, 128=a27

128=a×128

a=128128=1

a=1


Question 5

If the sum of first six terms of any G.P. is equal to 9 times the sum of the first three terms, then find the common ratio of the G.P.

Sol :

Sum of first 6 terms of a G.P. = 9x The of first 3 terms

Let a be the first term and r be the common ratio

S6=9×a3
Sn=a(rn1)r1

S6=a(r61)r1 and S3=a(r31)r1

a(r61)r1=9×a(r31)r1

r61=9(r31)

r61r31=9

(r3+1)(r31)r31=9

r3+1=9

r3=91=8=(2)2

r=2

Common ratio =2


Question 6

A G.P. consists of an even number of terms. If the sum of all the terms is 3 times the sum of the odd terms, then find its common ratio.

Sol :

Let the G.P. be a, a,ar,ar2,ar2n1

These are 2n in number, which is an even number

A.T.Q.

a+ar+ar2++ar2n1

=3(a+ar2+ar4++ar2n2)

a(ar2n1)r1r=3[a(ar2n2)r1r]

[ Using Sn=alrlr]

1r2n=3[1r2n1+r]

1+r=3r=31r=2

Common ratio is 2


Question 7

(i) How many terms of the G.P. 3,32,33, are needed to give the sum 120?

(ii) How many terms of the G.P. 1, 4, 16, … must be taken to have their sum equal to 341?

Solution:

In G.P.

(i) 3,32,33

Sum =120, Here, a=3,r=323=3,r>1

Let number of terms in G.P. be n, then

Sn=a(rn1)r1=120

3(3n1)31=120

3(3n1)2=120

3n1=120×23=80

3n=80+1=81=34

n=4

Number of terms =4


(ii) G.P. is 1,4,16,..

Sum=341

Here, a=1,r=41=4,r>1

Let number of terms be n, then

Sn=a(rn1)r1=341

1(4n1)41=341

1(4n1)41=341

4n13=341

4n1=341×3=1023

4n=1023+1=1024=210=45

210242512225621282642322162824221

n=5

Number of terms =5


Question 8

How many terms of the GP. 1, √2 > 2, 2 √2 , … are required to give a sum of 1023( √2 + 1)?

Sol :

GP. 1, √2 > 2, 2 √2 , …

Sum = 1023 (√2 + 1)

Here, a = 1, r = √2 . r > 1

Let number of terms be n, then

Sn=a(rn1)r1=1023(2+1)

1[(2)n1]21=1023(2+1)

(2)n1=1023(2+1)(21)

(2)n1=1023[(2)2(1)2]

(2)n1=1023(21)=1023

(2)n=1023+1=1024=210 or (2)20

210242512225621282642322162824221

Comparing, we get

n=20


Question 9

How many terms of the  2913+12+  will make the sum 5572 ?

Sol :

G.P. is 2913+12+

sum 5572

Here, a=29,r=13×92=32,r<1

Let number of terms be n

Sn=a(1rn)1r=5572

29[1(32)n]1+32=5572

1(32)n=5572×52×92=27532

1(1)n(32)n=27532

1+1(32)n=27532

(32)n=275321=2753232=24332

=(32)5

Comparing, we get

n=5

Number of terms =5


Question 10

The 2nd and 5th terms of a geometric series are 12 and sum 116  respectively. Find the sum of the series upto 8 terms.

Sol :
In a GP
a2=12 and a5=116

Let a be the first term and r be the common ratio

a2=arn1=ar21=ar=12...(i)

a5=ar51=ar4=116..(ii)

Dividing (ii) by (i),

r3=116÷(12)=116×21=18

=(12)3

r=12

and ar=12a×(12)=12

a=12×21=1

a=1, and r=12


Now,S8=a(1rn)1r

=1[1(12)8]1+12

=1125632

=255256×23=510768=85128


Question 11

The first term of a G.P. is 27 and 8th term is 181 . Find the sum of its first 10 terms.

Sol :

In a G.P.

First term (a) = 27

a8=81

Let r be the common ratio, then

arn1=ar81=ar7=181

27r7=181

r7=181×27=12187=1(3)7

r=13


Now, S10=a(1rn)1r

=27[1(13)10]113

=27[11310]313

=27×32[11310]=812[11310]


Question 12

Find the first term of the G.P. whose common ratio is 3, last term is 486 and the sum of whose terms is 728

Sol :
Common ratio of a G.P. = 3
and last term = 486
and sum of terms = 728

Sn=a(rn1)r1=a(3n1)31

=a(3n1)2=728

a(3n1)=728×2=1456...(i)
l=486
arn1=a3n1=486

a[3n3]=486

a3n=486×3=1458..(ii0)

But a(3n1)=1456 [From (i)]

a3na=1456

1458a=1456 [From (ii)]

a=14581456=2

Hence first term =2


Question 13

In a G.P. the first term is 7, the last term is 448, and the sum is 889. Find the common ratio.

Sol :

In a GP.

First term (a) = 7, last term (l) = 448

and sum = 889

Let r be the common ratio, then

l=arn17rn1=448
rn1=4487=64 ..(i)

and sum =a(rn1)r1=889

7(rn1)r1=889

rn1r1=8897=127..(ii)

From (i)

rnr=64rn=64r

From (ii) 64r1r1=127

64r1=127r127

127r64r=1+127

63r=126

r=12663=2

Hence common ratio =2


Question 14

Find the third term of a G.P. whose common ratio is 3 and the sum of whose first seven terms is 2186.

Sol :

In a G.P.

Common ratio = 3

S7=2186=a(rn1)r1

2186=a(3n1)31

2186=a(3n1)2

4372=a(371)

4372=a(21871)

4372=2186a

a=43722186=2

a3=ar31=ar2=2×32=2×9=18


Question 15

If the first term of a G.P. is 5 and the sum of first three terms is 315, find the common ratio.

Sol :

In a G.P.

First term (a) = 5

S3=315=a(r31)r1=5(r31)r1

r31r1=315×5=3125

(r1)(r2+r+1)r1=3125

r2+r+1=3125

25r2+25r+25=31

25r2+25r6=0

25r2+30r5r6=0

5r(5r+6)1(5r+6)=0

(5r+6)(5r1)=0

Either 5r+6=0, then r=65

or 5r-1=0, then r=15

Hence common ratio =15 or 65


Question 16

The sum of first three terms of a GP. is to the sum of first six terms as 125 : 152. Find the common ratio of the GP.

Sol :

S3÷S6=125:152

Let r be the common ratio and a be the first number, then

a(r31)r1:a(r61)r1=125:152
(r31):(r61)=125:152
(r31):(r3+1)(r31)=125:152
1:(r3+1)=125:152
(r3+1)×125=152×1
125r3+125=152
125r3=152125=27

r3=27125=(35)3

r=35

Common ratio =35


Question 17

Find the sum of the products of the corresponding terms of the geometric progression 2, 4, 8, 16, 32 and 128, 32, 8, 2, 12

Sol :

Sum of the product of corresponding terms of the G.M.s

2,4,8,16,32 and 128,32,8,2,12

=2×128+4×32+8×8+16×2+32×12

=256+128+64+32+16

Here, a=256,r=128256=12,n=5

S5=a(1rn)1r

=256(112)5112

S5=256(1132)12=256×21(3132)

S5=8×2×31=496


Question 18

Evaluate 50n=1(2n1)

Sol :

50n=1(2n1)

Here n = 1, 2, 3,….,50

Sn=211+221+231+2412501

Sn=(21+22+23+24250)1×50

 Sn=2+4+8+1625050

Sn=a(rn1)r150=2(2501)2150

 Sn=2×250250

 Sn=25152


Question 19

Find the sum of n terms of a series whose mth term is 2m+2m.

Sol :

am=2m+2m

a1=21+2×1=2+2

a2=22+2×2=4+4

a3=23+2×3=8+6

a4=24+2×4=16+8

an=2n+2×n

Sum =(2+4+8+16 n term )+(2+4+6+8 n term )

Sn=a(rn1)r1+n2[2a+(n1)d]

=2(2n1)21+n2[2×2+(n1)×2]

{a=2,r=2,d=2}

=2(2n1)1+n2[4+2(n1)]

=2(2n1)+n(2+n1)

=2(2n1)+n(n+1)


Question 20

Sum the series

x(x+y)+x2(x2+y2)+x3(x3+y3)… to n terms.

Sol :

Given

Sn=x(x+y)+x2(x2+y2)+x3(x3+y3)n terms

Sn=x2+xy+x4+x2y2+x6+x3y3+n terms

Sn=x2+x4+x6+n terms +xy+x2y2+x3y3+

Sn=x2[(x2)n1]x21+xy[(xy)n1]xy1

{ In the first G.P. a=x2,r=x2 In second G.P. a=xy,r=xy}

Sn=x2(x2n1)x21+xy[(xy)n1]xy1


Question 21

Find the sum of the series

1+(1+x)+(1+x+x2)+ to n terms, x1

Sol :

1+(1+x)+(1+x+x2)+ n terms, x1

Multiply and divide by (1 – x)

(1x)(1x)+(1x)(1+x)(1x)+(1x)(1+x+x2)(1x)+ upto n terms

1(1x)[(1x)+(1+x)2+(1+x3)+ upto n terms

1(1x)[(1+1+1+ upto n terms )(x+x2+x3+ upto n terms )

1(1x)[nx(1xn)(1x)]

[ Using Sn=a(1rn)1r as r<1]

1(1x)[n(1x)x(1xn)(1x)]

1(1x)2[n(1x)x(1xn)]


Question 22

Find the sum of the following series to n terms:

(i) 7 + 77 + 777 + …

(ii) 8 + 88 + 888 + …

(iii) 0.5 + 0.55 + 0.555 + …

Sol :

(i) 7 + 77 + 777 + … n terms

= 7[1 + 11 + 111 + … n terms]

=79[9+99+999+n terms ]

=79[(101)+(1001)+(10001)+ n terms]

=79[10+100+1000+n terms (1+1+1... n terms ]

=79[10(10n1)101n]

=79[10×10n109n]

=781[10n+1109n]

=781[10n+19n10]


(ii) 8+88+888+n terms

=8[1+11+111+n terms ]

=89[9+99+999+n terms ]

=89[(101)+(1001)+(10001)+...n terms]

=89[10+100+1000+n terms 1×n]

=89[10(10n1)101n]

=89[10n+1109n]

=881[10n+1109n]

=881[10n+19n10]


(iii) 0.5+0.55+0.555+... n terms

=5[0.1+0.11+0.111+... n terms ]

=59[0.9+0.99+0.999+ n terms ]

=59[(10.1)+(10.01)+(10.001)+... n terms)

=59[1+1+1+n terms (0.1+0.01+0.001)+...n terms]

=59[n0.1[1(0.1)n10.1]

=59[n19(1110n)]

=581[9n1+110n]

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