# Balbharati solutions for Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board chapter 4 - Sequences and Series [Latest edition]

## Chapter 4: Sequences and Series

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Miscellaneous Exercise 4
Exercise 4.1 [Pages 50 - 51]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board Chapter 4 Sequences and Series Exercise 4.1 [Pages 50 - 51]

Exercise 4.1 | Q 1. (i) | Page 50

Verify whether the following sequence is G.P. If so, write tn:

2, 6, 18, 54, ...

Exercise 4.1 | Q 1. (ii) | Page 50

Verify whether the following sequence is G.P. If so, write tn:

1, – 5, 25, – 125, ...

Exercise 4.1 | Q 1. (iii) | Page 50

Verify whether the following sequence is G.P. If so, write tn:

sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5)), ...

Exercise 4.1 | Q 1. (iv) | Page 50

Verify whether the following sequence is G.P. If so, write tn:

3, 4, 5, 6, ...

Exercise 4.1 | Q 1. (v) | Page 50

Verify whether the following sequence is G.P. If so, write tn:

7, 14, 21, 28, ...

Exercise 4.1 | Q 2. (i) | Page 50

For the G.P., if r = 1/3, a = 9, find t7.

Exercise 4.1 | Q 2. (ii) | Page 50

For the G.P., if a = 7/243, "r" = 1/3, find t3.

Exercise 4.1 | Q 2. (iii) | Page 50

For the G.P., if a = 7, r = – 3, find t6.

Exercise 4.1 | Q 2. (iv) | Page 50

For the G.P., if a = 2/3, t6 = 162, find r.

Exercise 4.1 | Q 3 | Page 50

Which term of the G. P. 5, 25, 125, 625, … is 510?

Exercise 4.1 | Q 4 | Page 50

For what values of x, 4/3, x, 4/27 are in G.P.?

Exercise 4.1 | Q 5 | Page 50

If for a sequence, tn = (5^("n" - 3))/(2^("n" - 3), shothat the sequence is a G. P. Find its first term and the common ratio.

Exercise 4.1 | Q 6 | Page 51

Find three numbers in G. P. such that their sum is 21 and sum of their squares is 189.

Exercise 4.1 | Q 7 | Page 51

Find four numbers in G. P. such that sum of the middle two numbers is 10/3 and their product is 1.

Exercise 4.1 | Q 8 | Page 51

Find five numbers in G. P. such that their product is 1024 and fifth term is square of the third term.

Exercise 4.1 | Q 9 | Page 51

The fifth term of a G. P. is x, eighth term of the G. P. is y and eleventh term of the G. P. is z. Verify whether y2 = xz.

Exercise 4.1 | Q 10 | Page 51

If p, q, r, s are in G. P., show that p + q, q + r, r + s are also in G. P.

Exercise 4.2 [Pages 54 - 55]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board Chapter 4 Sequences and Series Exercise 4.2 [Pages 54 - 55]

Exercise 4.2 | Q 1. (i) | Page 54

For the following G.P.'s, find Sn: 3, 6, 12, 24, ...

Exercise 4.2 | Q 1. (ii) | Page 54

For the following G.P.'s, find Sn: p, q, "q"^2/"p", "q"^3/"p"^2, ...

Exercise 4.2 | Q 2. (i) | Page 54

For a G.P., if a = 2, r = -2/3, find S6.

Exercise 4.2 | Q 2. (ii) | Page 54

For a G.P., if S5 = 1023, r = 4, find a.

Exercise 4.2 | Q 3. (i) | Page 54

For a G.P., if a = 2, r = 3, Sn = 242, find n.

Exercise 4.2 | Q 3. (ii) | Page 54

For a G.P., if sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r.

Exercise 4.2 | Q 4. (i) | Page 55

For a G.P., if t3 = 20, t6 = 160, find S7.

Exercise 4.2 | Q 4. (ii) | Page 55

For a G.P., if t4 = 16, t9 = 512, find S10.

Exercise 4.2 | Q 5. (i) | Page 55

Find the sum to n terms: 3 + 33 + 333 + 3333 + ...

Exercise 4.2 | Q 5. (ii) | Page 55

Find the sum to n terms: 8 + 88 + 888 + 8888 + …

Exercise 4.2 | Q 6. (i) | Page 55

Find the sum to n term: 0.4 + 0.44 + 0.444 + …

Exercise 4.2 | Q 6. (ii) | Page 55

Find the sum to n terms: 0.7 + 0.77 + 0.777 + ...

Exercise 4.2 | Q 7. (i) | Page 55

Find the nth terms of the sequences: 0.5, 0.55, 0.555, …

Exercise 4.2 | Q 7. (ii) | Page 55

Find the nth terms of the sequences:  0.2, 0.22, 0.222, …

Exercise 4.2 | Q 8 | Page 55

For a sequence, if Sn = 2 (3n – 1), find the nth term, hence show that the sequence is a G.P.

Exercise 4.2 | Q 9 | Page 55

If S, P, R are the sum, product and sum of the reciprocals of n terms of a G.P. respectively, then verify that ("S"/"R")^"n" = "P"^2.

Exercise 4.2 | Q 10 | Page 55

If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.

Exercise 4.3 [Pages 56 - 57]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board Chapter 4 Sequences and Series Exercise 4.3 [Pages 56 - 57]

Exercise 4.3 | Q 1. (i) | Page 56

Determine whether the sum to infinity of the following G.P’.s exist. If exists, find it:

1/2, 1/4, 1/8, 1/16, ...

Exercise 4.3 | Q 1. (ii) | Page 56

Determine whether the sum to infinity of the following G.P’.s exist. If exists, find it:

2, 4/3, 8/9, 16/27, ...

Exercise 4.3 | Q 1. (iii) | Page 56

Determine whether the sum to infinity of the following G.P’.s exist. If exists, find it:

-3, 1, (-1)/3, 1/9, ...

Exercise 4.3 | Q 1. (iv) | Page 57

Determine whether the sum to infinity of the following G.P’.s exist. If exists, find it:

1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...

Exercise 4.3 | Q 2. (i) | Page 57

Express the following recurring decimal as a rational number:

0.bar32

Exercise 4.3 | Q 2. (ii) | Page 57

Express the following recurring decimals as a rational number:

3.dot5

Exercise 4.3 | Q 2. (iii) | Page 57

Express the following recurring decimals as a rational number:

4.bar18

Exercise 4.3 | Q 2. (iv) | Page 57

Express the following recurring decimals as a rational number:

0.3bar45

Exercise 4.3 | Q 2. (v) | Page 57

Express the following recurring decimals as a rational number: 3.4bar56

Exercise 4.3 | Q 3 | Page 57

If the common ratio of a G.P. is 2/3 and sum of its terms to infinity is 12. Find the first term.

Exercise 4.3 | Q 4 | Page 57

If the first term of a G.P. is 16 and sum of its terms to infinity is 176/5, find the common ratio.

Exercise 4.3 | Q 5 | Page 57

The sum of the terms of an infinite G.P. is 5 and the sum of the squares of those terms is 15. Find the G.P.

Exercise 4.4 [Pages 60 - 61]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board Chapter 4 Sequences and Series Exercise 4.4 [Pages 60 - 61]

Exercise 4.4 | Q 1. (i) | Page 60

Verify whether the following sequence is H.P.:

1/3, 1/5, 1/7, 1/9, ...

Exercise 4.4 | Q 1. (ii) | Page 60

Verify whether the following sequence is H.P.:

1/3, 1/6, 1/9, 1/12, ...

Exercise 4.4 | Q 1. (iii) | Page 60

Verify whether the following sequence is H.P.:

1/7, 1/9, 1/11, 1/13, 1/15, ...

Exercise 4.4 | Q 2. (i) | Page 60

Find the nth term and hence find the 8th term of the following H.P.s:

1/2, 1/5, 1/8, 1/11, ...

Exercise 4.4 | Q 2. (ii) | Page 60

Find the nth term and hence find the 8th term of the following H.P.s:

1/4, 1/6, 1/8, 1/10, ...

Exercise 4.4 | Q 2. (iii) | Page 60

Find the nth term and hence find the 8th term of the following H.P.s:

1/5, 1/10, 1/15, 1/20, ...

Exercise 4.4 | Q 3 | Page 60

Find A.M. of two positive numbers whose G.M. and H.M. are 4 and 16/5.

Exercise 4.4 | Q 4 | Page 60

Find H.M. of two positive numbers whose A.M. and G.M. are 15/2 and 6.

Exercise 4.4 | Q 5 | Page 60

Find G.M. of two positive numbers whose A.M. and H.M. are 75 and 48.

Exercise 4.4 | Q 6 | Page 60

Insert two numbers between 1/7 and 1/13 so that the resulting sequence is a H.P.

Exercise 4.4 | Q 7 | Page 60

Insert two numbers between 1 and – 27 so that the resulting sequence is a G.P.

Exercise 4.4 | Q 8 | Page 60

Find two numbers whose A.M. exceeds their G.M. by 1/2 and their H.M. by 25/26.

Exercise 4.4 | Q 9 | Page 61

Find two numbers whose A.M. exceeds G.M. by 7 and their H.M. by 63/5.

Exercise 4.5 [Page 63]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board Chapter 4 Sequences and Series Exercise 4.5 [Page 63]

Exercise 4.5 | Q 1 | Page 63

Find the sum sum_("r" = 1)^"n"("r" + 1)(2"r" - 1).

Exercise 4.5 | Q 2 | Page 63

Find $\displaystyle\sum_{r=1}^{n} (3r^2 - 2r + 1)$.

Exercise 4.5 | Q 3 | Page 63

Find $\displaystyle\sum_{r=1}^{n}\frac{1 + 2 + 3 + ...+ r}{r}$

Exercise 4.5 | Q 4 | Page 63

Find sum_("r" = 1)^"n" (1^3 + 2^3 + ... + "r"^3)/("r"("r" + 1).

Exercise 4.5 | Q 5 | Page 63

Find the sum 5 x 7 + 7 x 9 + 9 x 11 x 13 + ... upto n terms.

Exercise 4.5 | Q 6 | Page 63

Find the sum 22 + 42 + 62 + 82 + ... upto n terms.

Exercise 4.5 | Q 7 | Page 63

Find (702 – 692) + (682 – 672) + ... + (22 – 12)

Exercise 4.5 | Q 8 | Page 63

Find the sum 1 x 3 x 5 + 3 x 5 x 7 + 5 x 7 x 9 + ... + (2n – 1) (2n + 1) (2n + 3)

Exercise 4.5 | Q 9 | Page 63

Find n, if (1 xx 2 + 2 xx 3 + 3 xx 4 + 4 xx 5 + ... "upto  n terms")/(1 + 2 + 3 + 4 + ... "upto n terms")= 100/3.

Exercise 4.5 | Q 10 | Page 63

If S1, S2 and S3 are the sums of first n natural numbers, their squares and their cubes respectively, then show that: 9S22 = S3(1 + 8S1).

Miscellaneous Exercise 4 [Pages 63 - 64]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board Chapter 4 Sequences and Series Miscellaneous Exercise 4 [Pages 63 - 64]

Miscellaneous Exercise 4 | Q 1 | Page 63

In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term.

Miscellaneous Exercise 4 | Q 2 | Page 63

For a G.P. a = 4/3 and "t"_7 = 243/1024, find the value of r.

Miscellaneous Exercise 4 | Q 3 | Page 64

For a sequence, if tn = (5^("n" - 2))/(7^("n" - 3)), verify whether the sequence is a G.P. If it is a G.P., find its  first term and the common ratio.

Miscellaneous Exercise 4 | Q 4 | Page 64

Find three numbers in G.P., such that their sum is 35 and their product is 1000.

Miscellaneous Exercise 4 | Q 5 | Page 64

Find four numbers in G. P. such that sum of the middle two numbers is 10/3 and their product is 1.

Miscellaneous Exercise 4 | Q 6 | Page 64

Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.

Miscellaneous Exercise 4 | Q 7 | Page 64

For a sequence Sn = 4(7n – 1), verify whether the sequence is a G.P.

Miscellaneous Exercise 4 | Q 8 | Page 64

Find 2 + 22 + 222 + 2222 + … upto n terms.

Miscellaneous Exercise 4 | Q 9 | Page 64

Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, …

Miscellaneous Exercise 4 | Q 10 | Page 64

Find $\displaystyle\sum_{r=1}^{n}(5r^2 + 4r - 3)$.

Miscellaneous Exercise 4 | Q 11 | Page 64

Find $\displaystyle\sum_{r=1}^{n}r(r-3)(r-2)$.

Miscellaneous Exercise 4 | Q 12 | Page 64

Find $\displaystyle\sum_{r=1}^{n}\frac{1^2 + 2^2 + 3^2+...+r^2}{2r + 1}$

Miscellaneous Exercise 4 | Q 13 | Page 64

Find $\displaystyle\sum_{r=1}^{n}\frac{1^3 + 2^3 + 3^3 +...+r^3}{(r + 1)^2}$

Miscellaneous Exercise 4 | Q 14 | Page 64

Find 2 x + 6 + 4 x 9 + 6 x 12 + ... upto n terms.

Miscellaneous Exercise 4 | Q 15 | Page 64

Find 122 + 132 + 142 + 152 + … + 202.

Miscellaneous Exercise 4 | Q 16 | Page 64

Find (502 – 492) + (482 –472) + (462 – 452) + .. + (22 –12).

Miscellaneous Exercise 4 | Q 17 | Page 64

In a G.P., if t2 = 7, t4 = 1575, find r.

Miscellaneous Exercise 4 | Q 18 | Page 64

Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.

Miscellaneous Exercise 4 | Q 19 | Page 64

If pth, qth and rth terms of a G.P. are x, y, z respectively, find the value of xq – r .yr – p .zp – q.

## Chapter 4: Sequences and Series

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Miscellaneous Exercise 4

## Balbharati solutions for Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board chapter 4 - Sequences and Series

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