NCERT Solutions Class 11 Chapter 8 Sequences And Series Miscellaneous Exercise (original) (raw)
Last Updated : 22 Apr, 2024
Question 1. If f is a function satisfying f(x + y) = f(x) f(y) for all x, y ∈ N such that, f(1) = 3 and \sum_{x=1}^{x=n} f(x) = 120, find the value of n.**
**Solution:
_We are given that,
_f(x + y) = f(x) × f(y) for all x, y ∈ N … (1)
_f(1) = 3
_Putting x = y = 1 in (1), we have
_f(1 + 1) = f(2) = f(1) f(1) = 3 × 3 = 9
_Similarly,
_f(1+1+1) = f(3) = f(1+2) = f(1) f(2) = 3 × 9 = 27
_And, f(4) = f(1+3) = f(1) f(3) = 3 × 27 = 81
_We have, f(1), f(2), f(3).., i.e., 3, 9, 27, …,
_This series forms a G.P. with the first term(a)=3 and common ratio(r)=3.
_We know that sum of terms in G.P is given by,
_S n _= \frac{a(r^n –1)}{r–1}
And also it’s given that the sum of terms of the function is 120.
_=> 120 = \frac{3(3^n–1)}{3–1}
_=> 3 n_–__1 = 80
_=> n=4
**Therefore, number of terms is 4.
**Question 2. The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
**Solution:
Let the total number of terms be n.
We know that, sum of terms of a G.P. is,
Sn = \frac{a(r^n –1)}{r–1}
Also, we know that the first term(a) is 5 and common ratio(r) is 2.
=> 315 = \frac{5(2^n–1)}{2–1}
=> 2n–1 = 63
=> n=6
The last term of the G.P = 6th term = ar6-1 = (5)(2)5 = 160
**Therefore, last term of the G.P. is 160 and the number of terms is 6.
**Question 3. The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
**Solution:
_Let the first term and common ratio of the G.P. be a and r respectively.
_We have, a = 1. So,
_a 3 _= ar 2 = r2 …. (1)
a 5 = ar_ _4_= r4.. _…. (2)
_According to the question, we have
_a 3 _+ a 5 = 90
_From (1) and (2),
_r 2 _+ r 4 = 90
r 4 + r2 – 9__0 = 0
_Solving for r 2 , we get,
r^2 = \frac{-1+\sqrt{(1+360)}}{2} = \frac{-1±19}{2}
_r 2 = –__10 or r2 = 9
_Taking only real roots, we get,
_r = ±3
**Therefore, the common ratio of the G.P. is ±3.
**Question 4. The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.
**Solution:
_Suppose the three numbers in G.P. are a, ar, and ar 2 .
_According to question, we have
_=> a + ar + ar 2 _= 56
_=> a (1 + r + r 2 ) = 56
_=> a = \frac{56}{(1+r+r^2)} _…. (1)
_Also, we are given that,
_a – 1, ar – 7, ar 2 _– 21 forms an A.P.
_which implies,
_=> (ar – 7) – (a – 1) = (ar 2 _– 21) – (ar – 7)
_=> ar – a – 6 = ar 2 – ar – 14
_=> ar 2 _– 2ar + a = 8
_=> a (r – 1) 2 _= 8 ….. (2)
_Putting value of a from (1) in (2), we get,
=> \frac{56}{(1+r+r^2)}_(r_– _1) 2 = 8
_=> 7(r 2 _– 2r + 1) = 1 + r + r 2
_=> 6r 2 _– 15r + 6 = 0
_=> (6r – 3) (r – 2) = 0
_=> r = 2, 1/2
_Now if r = 2, then a = 8 and the three numbers in G.P. are 8, 16, and 32.
_If r = 1/2, then a = 32 and the three numbers in G.P. are 32, 16, and 8.
**Therefore, in both the cases, the three numbers are 8, 16, and 32.
**Question 5. A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
**Solution:
_Suppose the terms in the G.P. are a 1, a 2 , a 3 , a 4 , … a 2n .
_Number of terms = 2n
_According to the question, we have
_=> a 1 _+ a 2 + a 3 _+ …+ a 2n = 5 [a 1 + a 3 + … + a 2n–1 ]
_=> a 1 _+ a 2 + a 3 + … + a 2n – 5 [a 1 + a 3 + … + a 2n–1 ] = 0
_=> a 2 _+ a 4 + … + a 2n = 4 [a 1 + a 3 + … + a 2n – 1 ] …. (1)
_Now, let the terms of our G.P. be a, ar, ar 2 , ar 3 , …..
_Using sum of terms of a G.P., equation (1) becomes,
\frac{ar(r^n-1)}{r-1} = \frac{4a(r^n-1)}{r-1}
_After solving we get,
_ar = 4a
_r = 4
**Thus, the common ratio of the G.P. is 4.
**Question 6. If, \frac{a+bx}{a–bx} = \frac{b+cx}{b–cx} = \frac{c+dx}{c–dx} ****, then show that a, b, c and d are in G.P.**
**Solution:
_We are given,
\frac{a+bx}{a–bx} = \frac{b+cx}{b–cx}
_On cross multiplication, we have,
_=> (a+bx)(b-cx) = (b+cx)(a-bx)
_=> ab–acx+b 2 x–bcx 2 = ab–b 2 x+acx–bcx 2
_=> 2b 2 x = 2acx
_=> b 2 _= ac
_=> b/a = c/b ….. (1)
_We are also given,
\frac{b+cx}{b–cx} = \frac{c+dx}{c–dx}
On cross multiplication, we have,
_=> (b+cx)(c_−__dx) = (c+dx)(b_−__cx)
_=> bc_−__bdx+c 2 x-cdx 2 = bc+bdx_−__c 2 x_−__cdx 2
_=> 2c 2 x = 2bdx
_=> c 2 _= bd
_=> c/d = d/c ….. (2)
_From (1) and (2), we get
_b/a = c/b = d/c
**Therefore, a, b, c and d are in G.P.
**Question 7. Let S be the sum, P the product, and R the sum of reciprocals of n terms in a G.P. Prove that P 2 R n = S n .
**Solution:
_Suppose the terms in the G.P. are a, ar, ar 2 , ar 3 , … ar n – 1 …
_According to the question, we have,
_S = a(r n_−__1)/(r_−__1) ….. (1)
_P = a n _× r 1+2+....+n-1
_= a n r _n(n-1)/2
_R = 1/a + 1/ar + 1/ar 2 _+ …. + 1/ar n-1
_= \left(\frac{ r^{n−1}}{r-1}\right) \left(\frac{1}{ar^{n-1}}\right)
_So, P 2 R n = a 2n r _n(n-1) _= \frac{(r^n-1)^{n}}{a^nr^{n(n-1)}(r-1)^n}
_= \frac{a^n(r^n-1)^n}{(r-1)^n}
_= \left[\frac{a(r^n-1)}{r-1}\right]^{n}
_From (1), we get,
_P 2 R n = S n
**Hence, proved
**Question 8. If a, b, c, d are in G.P, prove that (a n + b n ), (b n + c n ), (c n + d n ) are in G.P.
**Solution:
_We are given, a, b, c and d are in G.P.
_Therefore, we have
_b 2 _= ac … (1)
_c 2 _= bd … (2)
_ad = bc … (3)
_We need to prove (a n _+ b n ), (b n + c n ), (c n + d n ) are in G.P. i.e.,
_=> (b n + c n ) 2 = (a n + b n ) (c n + d n )
_Solving L.H.S., we get
_= b 2n + 2b n c n + c 2n
_= (b 2)n _+ 2b n c n + (c 2 ) n
_= (ac) n _+ 2b n c n + (bd) n [From (1) and (2)]
_= a n c n + b n c n + b n c n + b n d n
_= a n c n + b n c n + a n d n + b n d n [From (3)]
_= c n (a n + b n ) + d n (a n + b n )
_= (a n _+ b n ) (c n + d n )
_= R.H.S.
_Therefore, (a n _+ b n ), (b n + c n ), and (c n + d n ) are in G.P.
**Hence, proved.
**Question 9. **If a and b are the roots of x 2 – 3x + p = 0 and c, d are roots of x 2 **– 12x + q = 0, where a, b, c, d, form a G.P. Prove that
****(q + p):(q – p) = 17:15.**
**Solution:
_We are given that a, b, c, d are in G.P. Let's suppose the common ratio is r.
_So, b=ar, c=ar 2 and d=ar 3
_Now a and b are the roots of x 2 _– 3x + p = 0.
Sum of roots = a + b = 3
_=> a + ar = 3
_=> a(1+r) = 3 ….. (1)
Product of roots = ab = p
_=> a(ar) = p
_=> a 2 r = p ….. (2)
_And c, d are the roots of x 2 _− 12x + q = 0
_Sum of roots = c + d = 12
_=> ar 2 _+ ar 3 = 12
_=> ar 2 (1+r) = 12 ….. (3)
_Product of roots = cd = q
_=> ar 2 (ar 3 ) = q
_=> a 2 r 5 = q ….. (4)
_Dividing equation (3) by (1),we get,
_=> \frac{ar^2(1+r)}{a(1+r)} = \frac{12}{3}
_=> r 2 _= 4
_=> r = ±2
**When r=2, from (1), we get,
_=> a(1+2) = 3
_=> a = 1
_Putting a=1 and r=2 in (2),
_=> p = (1) 2 (2) = 2
_From (4) we get,
_q = (1) 2 (2) 5 = 32
_Now L.H.S. = \frac{q+p}{q−p} = \frac{32+2}{32−2} = \frac{34}{30} _= \frac{17}{15}
_= R.H.S.
**When r=−2, from (1), we get,
_=> a(1−2) = 3
_=> a = −3
_Putting a=−3 and r=−2 in (2),
_p = (−3) 2 (−2) = −18
_From (4) we get,
_q = (−3) 2 (−2) 5 _= −288
_Now L.H.S. = \frac{q+p}{q−p} = \frac{−288+(−18)}{−288−(−18)} = \frac{−306}{−270} = \frac{17}{15}
_= R.H.S.
**Hence, proved.
**Question 10. The ratio of the A.M and G.M. of two positive numbers a and b, is m: n. Show that a:b = (m+√(m 2 −n 2 )):(m-√(m 2 -n 2 )).
**Solution:
_We are given two numbers a and b. Therefore,
_A.M = (a + b)/2 and G.M. = \sqrt{(ab)}
_It's given that, \frac{a+b}{2\sqrt{(ab)}} = \frac{m}{n}
_Applying Componendo and Dividendo on both sides, we get,
_=> \frac{a+b+2\sqrt{(ab)}}{a+b−2\sqrt{(ab)}} = \frac{m+n}{m−n}
_=> \frac{(\sqrt{a}+\sqrt{b})^2}{(\sqrt{a}−\sqrt{b})^2} = \frac{m+n}{m−n}
_=> \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}−\sqrt{b}} = \frac{\sqrt{(m+n)}}{\sqrt{(m−n)}}
_By again applying Componendo and Dividendo on both sides, we get,
_=> \frac{(\sqrt{a}+\sqrt{b}+\sqrt{a}−\sqrt{b})}{(\sqrt{a}+\sqrt{b}−\sqrt{a}+\sqrt{b})} = \frac{\sqrt{(m+n)}+\sqrt{(m−n)}}{\sqrt{(m+n)}−\sqrt{(m−n)}}
_=> \frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{(m+n)}+\sqrt{(m−n)}}{\sqrt{(m+n)}−\sqrt{(m−n)}}
_Squaring both sides, we get,
__=>_\frac{a}{b} = \frac{(m+n)+(m-n)+2\sqrt{(m+n)}\sqrt{(m−n)}}{(m+n)+(m-n)-2\sqrt{(m+n)}\sqrt{(m−n)}}
_=> \frac{a}{b} = \frac{2m+2\sqrt{(m^2−n^2)}}{2m-2\sqrt{(m^2−n^2)}}
_=> \frac{a}{b} = \frac{m+\sqrt{(m^2−n^2)}}{m-\sqrt{(m^2−n^2)}}
**Hence, proved.
**Question 11. Find the sum of the following series up to n terms:
****(i) 5 + 55 + 555 + …**
**Solution:
_Let us take S n _= 5 + 55 + 555 + ….. up to n terms
_Multiplying and dividing by 9, we get
_= \frac{5}{9} _[9+99+999+...to n terms]
_= \frac{5}{9} _[(10–1)+(10 2 –1)+(10 3 –1)...to n terms]
_= \frac{5}{9} _[(10+10 2 +10 3 +....n terms) – (1+1+1+…..n terms)]
_= \frac{5}{9} \left[\frac{10(10^n–1)}{10–1}-n \right]
_= \frac{5}{9} \left[\frac{10(10^n-1)}{9}-n\right]
= \frac{50(10^n-1)}{81} – \frac{5n}{9}
****(ii) .6 + .66 + .666 + …**
**Solution:
_Let us take S n _= 0.6 + 0.66 + 0.666 + … up to n terms
_= 6 [0.1+0.11+0.111+...up to n terms]
_Multiplying and dividing by 9, we get
_= \frac{6}{9} [0.9+0.99+0.999+...up to n terms]
_= \frac{6}{9} \left[\left(1–\frac{1}{10} \right)+\left(1–\frac{1}{10^2}\right)+\left(1–\frac{1}{10^3}\right)+...n\hspace{0.1cm}terms\right]
_= \frac{2}{3} \left[(1+1+1+...n\hspace{0.1cm}terms) – \frac{1}{10}\left(1+\frac{1}{10}+\frac{1}{10^2} +...n\hspace{0.1cm}terms\right)\right]
_= \frac{2}{3} \left[n-\frac{1}{10}\left[\frac{1-\left(\frac{1}{10}\right)^n}{1-\frac{1}{10}}\right]\right]
__=_\frac{2}{3} __–_\frac{2(1–10^{-n})}{27}
**Question 12. Find the 20th term of the series 2 × 4 + 4 × 6 + 6 × 8 + … + n terms.
**Solution:
_We are given the series: 2 × 4 + 4 × 6 + 6 × 8 + … n terms
_nth term = a n _= 2n × (2n + 2) = 4n 2 _+ 4n
_Putting n=20, we would get the 20th term,
_a 20 _= 4(20) 2 _+ 4(20) = 1600 + 80 = 1680
**Therefore, the 20th term of the series is 1680.
**Question 13. A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?
**Solution:
_We are given that the farmer pays Rs 6000 in cash.
_So, the remaining unpaid amount = Rs 12000 – Rs 6000 = Rs 6000
_According to the question, the total interest to be paid is,
_Interest = 12% of 6000 + 12% of 5500 + 12% of 5000 + … + 12% of 500
_= 12% of (6000 + 5500 + 5000 + … + 500)
_Here the series **6000 + 5500 + 5000 + … + 500 _is an A.P. with the first term a=6000 and common difference d=–500.
_Let us take the number of terms of A.P. be n.
_We know n th _term of an A.P. is given by, **a **n **= a+(n**– 1)d
_=> 500 = 6000+(n-1)(–500)
_=> 5500 = 500n–500
_=> n = 12
_Now,
_The sum of the A.P = \frac{12}{2} [2(6000) + (12–1)(-500)]
_= 6 [1200_–__5500] = 39000
_Thus, the total interest to be paid = 12% of (6000 + 5500 + 5000 + … + 500)
_= 12% of 39000 = Rs 4680
**Therefore, the tractor will cost the farmer = (Rs 12000 + Rs 4680) = Rs 16680
**Question 14. Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in the **annual installment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?
**Solution:
_We are given that Shamshad Ali buys a scooter for Rs 22000 and pays Rs 4000 in cash.
_So, the remaining unpaid amount = Rs 22000 – Rs 4000 = Rs 18000
_According to the question, the total interest to be paid,
_Interest = 10% of 18000 + 10% of 17000 + 10% of 16000 + … + 10% of 1000
_= 10% of (18000 + 17000 + 16000 + … + 1000)
_Here, **18000, 17000, 16000 … 1000 _forms an A.P. with first term a=18000 and common difference d=–1000.
_Let’s take number of terms be n.
_So, 1000 = 18000 + (n – 1) (–1000)
_=> 17000 = 1000n–1000
_=> n = 18
_Now, the sum of the A.P. = \frac{18}{2} [2(18000) + (18–1)(-1000)]
_= 9 [36000–17000] = 9 [19000] = 171000
_Thus, the total interest to be paid = 10% of (18000 + 17000 + 16000 + … + 1000)
_= 10% of 171000 = Rs 17100
**Thus, the cost of scooter = (Rs 22000 + Rs 17100) = Rs 39100
**Question 15. A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paisa to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.
**Solution:
_The numbers of letters mailed forms a G.P. : 4, 4 2 , … 4 8 _where first term, a = 4 and common ratio, r = 4
_And the number of terms, n = 8
_The sum of n terms of a G.P. is given by, S n __=_\frac{a(r^n–1)}{r–1}
_= \frac{4(4^8–1)}{4–1}
_= \left(\frac{4}{3}\right) _(65536–1) = 4(21845) = 87380
_Now it's given that the cost to mail one letter is 50 paisa.
_Thus, Cost of mailing 87380 letters = Rs 87380 x (50/100) = Rs 43690
**Therefore, the amount spent on the postage when 8th set of letter is mailed will be Rs 43690.
**Question 16. A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.
**Solution:
_Given that the man deposited Rs 10000 in a bank at the rate of 5% simple interest annually.
_Thus, the interest in first year = (5/100) x Rs 10000 = Rs 500
_Now, Amount in 15th year = 10000 + (500+500+500+...14 times)
_= Rs 10000 + 14 × Rs 500
_= Rs 10000 + Rs 7000 = Rs 17000
_Amount in 20th year = 10000 + (500+500+500+...20 times)
_= Rs 10000 + 20 × Rs 500
_= Rs 10000 + Rs 10000 = Rs 20000
**Therefore, the amount in the 15th year is Rs 17000 and the total amount after 20 years will be Rs 20000.
**Question 17. A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
**Solution:
_We are given the cost of machine is Rs 15625.
_Also, given that the machine depreciates by 20% every year.
_Therefore, its value after every year is 80% of the original cost i.e., 4/5th of the original cost.
_So, the value at the end of 5 years = 15625 × (\frac{4}{5} × \frac{4}{5} ×...\hspace{0.1cm}5\hspace{0.1cm}times) = 5 × 1024 = 5120
**Thus, the value of the machine at the end of 5 years will be Rs 5120.
**Question 18: 150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.
**Solution:
Let us take x to be the number of days in which 150 workers finish the work.
According to the question, we have
150x = 150 + 146 + 142 + ….. (x + 8) terms
We know, 150, 146, 142, ….. (x + 8) terms is an A.P.
with first term (a) = 150,
Common difference (d) = –4 and number of terms (n) = (x + 8)
Now, we know sum of this series is equal to 150x .
=> 150x = \frac{(x+8)}{2} [2×150 + (x+8–1)×4]
=> 300x = (x+8)(300–4x–28)
=> 300x = 272x–4x2+2176–32x
=> 4x2+60x+–2176 = 0
=> x2+15x–544 = 0
=> x2+32x–17x-544 = 0
=> (x+32)(x–17) = 0
=> x = –32 or 17
Ignoring x = –32 as number of days is a positive quantity so, x = 17.
It took 8 more days to finish the work because workers were dropped out.
**Hence, the number of days in which the work was completed is 17+8 = 25.