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 +…
(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
=2(320−1)3−1=2(320−1)2=320−1
(ii) 1+√3+3+…10 terms
Here, a=1,r=√3,n=10,r>1
S10=a(rn−1)r−1=1[(√3)10−1]√3−1
=(√310−1)(√3+1)(√3−1)(√3+1)
(Rationalising the denominator)
=(35−1)(√3+1)3−1=(243−1)(√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(1−rn)1−r=1[1−(−23)6]1+23
=35(1−(−2)636)
=35[1−64729]=35[729−64729]
=35×665729=133243
(iv) 0.15,0.015,0.0015, ... upto 20 terms
Here, a=0.15,r=110,n=20
S20=a(1−rn)1−r (∵r<1)
=0.15[1−(110)20]1−110
=15×10100×9[1−11020]
=530[1−11020]=16[1−11020]
(v) 0.7+0.07+0.007+... 100 terms
Here, a=0.7,r=110,n=100,r<1
S100=a(1−rn)1−r
=0.7[1−(110)100]1−110
=7×1010×9(1−110100)=79(1−110100)
(vi) 1+23+49+…n terms, 5 terms
Here, a=1,r=23÷1=23,n=5,n,(r<1)
Sn=a(1−rn)1−r=1[1−(23)n]1−23
=1×31[1−(23)n]=3[1−(23)n]
and S5=3[1−(23)5]=3[1−32243]
=3[243−32243]=21181
(vii) √7,√21,3√¯7,…
Here, a=√7,r=√21√7=√7×√3√7=√3
r>1,n=n terms
Sn=a(rn−1)r−1⇒Sn=√7[(√3)n−1]√3−1
Rationalising the denominator.
⇒Sn=√7[(√3)n−1]√3−1×√3+1√3+1
=√7[(√3)n−1](√3+1)(√3)2−(1)2
=√7[(√3)n−1](√3+1)3−1
=√72[(√3)n−1](√3+1)
(viii) 1,−a,a2,−a3,…(a≠−1) upto n terms
Here, a=1, r=-a
Sn=a(1−rn)1−r=1[1−(−a)n]1−(−a)=1−(−a)n1+a
(ix) x3,x5,x7,…(x≠±1)
Here, a=x3,r=x2
∴Sn=a(1−rn)1−r=x3[1−(x2)n]1−x2 if r<1
=x3(1−x2n)1−x2
or Sn=a(rn−1)1−r=x3[(x2)n−1]x2−1
=x3(x2n−1)x2−1
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
=√2√3−1(243−1)=√2√3−1×242
=√2(√3+1)×242(√3−1)(√3+1)
(Rationalising the denominator)
=242(√6+√2)3−1=121(√6+√2)2
=121(√6+√2)
Question 3
Find the sum of the series 81−27+9…−127
Sol :
81−27+9…−127
Here, a=81,r=−2781=−13,l=−127,r<1
Sn=a−lr1−r
=81+127×−131+13
=81−18143
=6561−181×43
=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.
Sn=a(rn−1)r−1
⇒255=a(2n−1)2−1
⇒255=a(2n−1)
⇒a=2552n−1..(ii)
From (i) and (ii)
2552n−1=1282n−1
⇒255×2n−1=128(2n−1)
⇒255×2n−1=128×2n−128
255×2n2=128×2n−128
⇒255×2n=256×2n−256
⇒256×2n−255×2n=256
⇒2n=256=28
Comparing, we get
n=8
Now, 128=a⋅27
⇒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=a(r6−1)r−1 and S3=a(r3−1)r−1
∴a(r6−1)r−1=9×a(r3−1)r−1
⇒r6−1=9(r3−1)
⇒r6−1r3−1=9
⇒(r3+1)(r3−1)r3−1=9
⇒r3+1=9
⇒r3=9−1=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,…ar2n−1
These are 2n in number, which is an even number
A.T.Q.
=3(a+ar2+ar4+…+ar2n−2)
⇒a−(ar2n−1)r1−r=3[a−(ar2n−2)r1−r]
[∵ Using Sn=a−lrl−r]
⇒1−r2n=3[1−r2n1+r]
⇒1+r=3⇒r=3−1⇒r=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(rn−1)r−1=120
⇒3(3n−1)3−1=120
⇒3(3n−1)2=120
⇒3n−1=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(rn−1)r−1=341
⇒1(4n−1)4−1=341
⇒1(4n−1)4−1=341
⇒4n−13=341
⇒4n−1=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
⇒1[(√2)n−1]√2−1=1023(√2+1)
⇒(√2)n−1=1023(√2+1)(√2−1)
⇒(√2)n−1=1023[(√2)2−(1)2]
⇒(√2)n−1=1023(2−1)=1023
(√2)n=1023+1=1024=210 or (√2)20
210242512225621282642322162824221
Comparing, we get
n=20
Question 9
How many terms of the 29−13+12+… will make the sum 5572 ?
Sol :
G.P. is 29−13+12+…
sum 5572
Here, a=29,r=−13×92=−32,r<1
Let number of terms be n
∴Sn=a(1−rn)1−r=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=27532−1=275−3232=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.
Let a be the first term and r be the common ratio
a5=ar5−1=ar4=116..(ii)
Dividing (ii) by (i),
r3=116÷(−12)=116×−21=−18
=(−12)3
∴r=−12
and ar=−12⇒a×(−12)=−12
⇒a=−12×−21=1
a=1, and r=−12
Now,S8=a(1−rn)1−r
=1[1−(−12)8]1+12
=1−125632
=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.
In a G.P.
First term (a) = 27
Let r be the common ratio, then
⇒27r7=181
⇒r7=181×27=12187=1(3)7
∴r=13
Now, S10=a(1−rn)1−r
=27[1−(13)10]1−13
=27[1−1310]3−13
=27×32[1−1310]=812[1−1310]
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
⇒a[3n3]=486
⇒a3n=486×3=1458..(ii0)
But a(3n−1)=1456 [From (i)]
a3n−a=1456
1458−a=1456 [From (ii)]
∴a=1458−1456=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
and sum =a(rn−1)r−1=889
⇒7(rn−1)r−1=889
⇒rn−1r−1=889−7=127..(ii)
From (i)
rnr=64⇒rn=64r
From (ii) 64r−1r−1=127
⇒64r−1=127r−127
⇒127r−64r=−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
⇒2186=a(3n−1)2
⇒4372=a(37−1)
⇒4372=a(2187−1)
⇒4372=2186a
⇒a=43722186=2
a3=ar3−1=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(r3−1)r−1=5(r3−1)r−1
⇒(r−1)(r2+r+1)r−1=3125
r2+r+1=3125
⇒25r2+25r+25=31
⇒25r2+25r−6=0
⇒25r2+30r−5r−6=0
⇒5r(5r+6)−1(5r+6)=0
⇒(5r+6)(5r−1)=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
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
=2×128+4×32+8×8+16×2+32×12
=256+128+64+32+16
Here, a=256,r=128256=12,n=5
S5=a(1−rn)1−r
=256(1−12)51−12
S5=256(1−132)12=256×21(3132)
S5=8×2×31=496
Question 18
Evaluate ∑50n=1(2n−1)
Sol :
∑50n=1(2n−1)
Here n = 1, 2, 3,….,50
Sn=(21+22+23+24…250)−1×50
Sn=2+4+8+16…250−50
Sn=a(rn−1)r−1−50=2(250−1)2−1−50
Sn=2×250−2−50
Sn=251−52
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(rn−1)r−1+n2[2a+(n−1)d]
=2(2n−1)2−1+n2[2×2+(n−1)×2]
{∵a=2,r=2,d=2}
=2(2n−1)1+n2[4+2(n−1)]
=2(2n−1)+n(2+n−1)
=2(2n−1)+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)n−1]x2−1+xy[(xy)n−1]xy−1
{ In the first G.P. a=x2,r=x2 In second G.P. a=xy,r=xy}
Sn=x2(x2n−1)x2−1+xy[(xy)n−1]xy−1
Question 21
Find the sum of the series
Sol :
1+(1+x)+(1+x+x2)+… n terms, x≠1
Multiply and divide by (1 – x)
⇒1(1−x)[(1+1+1+… upto n terms )−(x+x2+x3+… upto n terms )
⇒1(1−x)[n−x(1−xn)(1−x)]
[∵ Using Sn=a(1−rn)1−r as r<1]
⇒1(1−x)[n(1−x)−x(1−xn)(1−x)]
⇒1(1−x)2[n(1−x)−x(1−xn)]
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[(10−1)+(100−1)+(1000−1)+… n terms]
=79[10+100+1000+…n terms −(1+1+1... n terms ]
=79[10(10n−1)10−1−n]
=79[10×10n−109−n]
=781[10n+1−10−9n]
=781[10n+1−9n−10]
(ii) 8+88+888+…n terms
=8[1+11+111+…n terms ]
=89[9+99+999+…n terms ]
=89[(10−1)+(100−1)+(1000−1)+...n terms]
=89[10+100+1000+…n terms −1×n]
=89[10(10n−1)10−1−n]
=89[10n+1−109−n]
=881[10n+1−10−9n]
=881[10n+1−9n−10]
(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[(1−0.1)+(1−0.01)+(1−0.001)+... n terms)
=59[1+1+1+…n terms −(0.1+0.01+0.001)+...n terms]
=59[n−0.1[1−(−0.1)n1−0.1]
=59[n−19(1−110n)]
=581[9n−1+110n]
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