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## Solutions for Chapter 4: Sequences and Series

Below listed, you can find solutions for Chapter 4 of Maharashtra State Board Balbharati for Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board.

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

**Verify whether the following sequence is G.P. If so, write t _{n}: **

2, 6, 18, 54, ...

**Verify whether the following sequence is G.P. If so, write t _{n}: **

1, – 5, 25, – 125, ...

**Verify whether the following sequence is G.P. If so, write t _{n}:**

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

**Verify whether the following sequence is G.P. If so, write t _{n}:**

3, 4, 5, 6, ...

**Verify whether the following sequence is G.P. If so, write t _{n}:**

7, 14, 21, 28, ...

For the G.P., if r = `1/3`, a = 9, find t_{7}.

For the G.P., if a = `7/243, "r" = 1/3`, find t_{3}.

For the G.P., if a = 7, r = – 3, find t_{6}.

For the G.P., if a = `2/3,` t_{6} = 162, find r.

Which term of the G. P. 5, 25, 125, 625, … is 5^{10}?

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

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

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

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

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

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 y^{2} = xz.

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

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

For the following G.P.'s, find S_{n}: 3, 6, 12, 24, ...

For the following G.P.'s, find S_{n}: p, q, `"q"^2/"p", "q"^3/"p"^2`, ...

For a G.P., if a = 2, r = `-2/3`, find S_{6}.

For a G.P., if S_{5} = 1023, r = 4, find a.

For a G.P., if a = 2, r = 3, S_{n} = 242, find n.

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

For a G.P., if t_{3} = 20, t_{6} = 160, find S_{7}.

For a G.P., if t_{4} = 16, t_{9} = 512, find S_{10}.

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

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

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

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

Find the n^{th} terms of the sequences: 0.5, 0.55, 0.555, …

Find the n^{th} terms of the sequences: 0.2, 0.22, 0.222, …

For a sequence, if S_{n} = 2 (3^{n} – 1), find the n^{th} term, hence show that the sequence is a G.P.

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`.

If S_{n}, S_{2n}, S_{3n} are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that S_{n} (S_{3n} – S_{2n}) = (S_{2n} – S_{n})^{2}.

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

**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`, ...

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

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

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

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

**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`, ...

**Express the following recurring decimal as a rational number: **

`0.bar32`

**Express the following recurring decimals as a rational number:**

`3.dot5`

**Express the following recurring decimals as a rational number: **

`4.bar18`

**Express the following recurring decimals as a rational number:**

`0.3bar45`

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

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

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

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.

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

**Verify whether the following sequence is H.P.: **

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

**Verify whether the following sequence is H.P.: **

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

**Verify whether the following sequence is H.P.: **

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

**Find the n ^{th} term and hence find the 8^{th} term of the following H.P.s: **

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

**Find the n ^{th} term and hence find the 8^{th} term of the following H.P.s: **

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

**Find the n ^{th} term and hence find the 8^{th} term of the following H.P.s: **

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

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

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

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

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

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

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

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

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

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

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

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

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

Find the sum 5 x 7 + 9 × 11 + 13 × 15 + --- upto n terms.

Find the sum 2^{2} + 4^{2} + 6^{2} + 8^{2} + ... upto n terms.

Find (70^{2} – 69^{2}) + (68^{2} – 67^{2}) + ... + (2^{2} – 1^{2})

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

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`.

If S_{1}, S_{2} and S_{3} are the sums of first n natural numbers, their squares and their cubes respectively, then show that: 9S_{2}^{2} = S_{3}(1 + 8S_{1}).

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

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

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

For a sequence, if t_{n} = `(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.

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

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

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

For a sequence S_{n} = 4(7^{n} – 1), verify whether the sequence is a G.P.

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

Find the n^{th} term of the sequence 0.6, 0.66, 0.666, 0.6666, …

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

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

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

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

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

Find 12^{2} + 13^{2} + 14^{2} + 15^{2} + … + 20^{2}.

Find (50^{2} – 49^{2}) + (48^{2} –47^{2}) + (46^{2} – 45^{2}) + .. + (2^{2} –1^{2}).

In a G.P., if t_{2} = 7, t_{4} = 1575, find r.

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

If p^{th}, q^{th} and r^{th} terms of a G.P. are x, y, z respectively, find the value of x^{q – r} .y^{r – p} .z^{p – q}.

## Solutions for Chapter 4: Sequences and Series

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

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Concepts covered in Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board chapter 4 Sequences and Series are Concept of Sequences, Geometric Progression (G.P.), General Term Or the nth Term of a G.P., Sum of the First n Terms of a G.P., Sum of Infinite Terms of a G. P., Recurring Decimals, Harmonic Progression (H. P.), Types of Means, Special Series (Sigma Notation).

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