Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) [English] 11 Standard Maharashtra State Board chapter 2 - Sequences and Series [Latest edition]

Check whether the following sequence is G.P. If so, write t_n.

2, 6, 18, 54, …

Check whether the following sequence is G.P. If so, write t_n.

1, –5, 25, –125 …

Check whether the following sequence is G.P. If so, write t_n.

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

Check whether the following sequence is G.P. If so, write t_n.

3, 4, 5, 6, …

Check 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 t₇

For the G.P. if a = `7/243`, r = 3 find t₆.

For the G.P. if r = − 3 and t₆ = 1701, find a.

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

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

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

If for a sequence, t_n = `(5^("n"-3))/(2^("n"-3))`, show that 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 a G.P. is y and eleventh term of a G.P. is z verify whether y² = xz

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

The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5^thhour?

A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6^th bounce? How high does the ball rebound on n^th bounce?

The numbers 3, x, and x + 6 form are in G.P. Find x

The numbers 3, x, and x + 6 form are in G.P. Find 20^th term.

The numbers 3, x, and x + 6 form are in G.P. Find n^th term

Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 3 years.

Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 10 years.

Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after n years.

The numbers x − 6, 2x and x² are in G.P. Find x

The numbers x − 6, 2x and x² are in G.P. Find 1^st term

The numbers x − 6, 2x and x² are in G.P. Find n^th term

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 the following G.P.s, find S_n

0.7, 0.07, 0.007, .....

For the following G.P.s, find S_n.

`sqrt(5)`, −5, `5sqrt(5)`, −25, ...

For a G.P. a = 2, r = `-2/3`, find S₆

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

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

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

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

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

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 terms 0.4 + 0.44 + 0.444 + ...

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

Find the sum to n terms of the sequence.

0.5, 0.05, 0.005, ...

Find the sum to n terms of the sequence.

0.2, 0.02, 0.002, ...

For a sequence, if S_n = 2(3ⁿ –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"` = P² 

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.

Find: `sum_("r" = 1)^10(3 xx 2^"r")`

Find: `sum_("r" = 1)^10 5 xx 3^"r"`

The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is ₹ 15 Lac. [Given: (1.05)⁵ = 1.28, (1.05)⁶ = 1.34]

If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)⁵ = 1.47]

Determine whether the sum to infinity of the following G.P.s exist, if exists find them:

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

Determine whether the sum to infinity of the following G.P.s exist, if exists find them:

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

Determine whether the sum to infinity of the following G.P.s exist, if exists find them:

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

Determine whether the sum to infinity of the following G.P.s exist, if exists find them:

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

Determine whether the sum to infinity of the following G.P.s exist, if exists find them:

9, 8.1, 7.29, ...

Express the following recurring decimal as a rational number:

`0.bar(7)`

Express the following recurring decimal as a rational number:

`2.bar(4)`

Express the following recurring decimal as a rational number:

`2.3bar(5)`

Express the following recurring decimal as a rational number:

`51.0bar(2)`

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

If the first term of the G.P. is 6 and its sum to infinity is `96/17` find the common ratio.

The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.

Find : `sum_("r" = 1)^oo 4(0.5)^"r"`

Find : `sum_("r" = 1)^oo (-1/3)^"r"`

Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"` 

Find : `sum_("n" = 1)^oo 0.4^"n"`

The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares

The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares

A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball

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, 1/12, 1/24, ...`

Verify whether the following sequence is H.P.

`5, 10/17, 10/32, 10/47, ...`

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` respectively.

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

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

Insert two numbers between `1/4` and `1/3` 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.

If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.

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

Find S_n of the following arithmetico - geometric sequence: 

2, 4x, 6x², 8x³, 10x⁴, …

Find S_n of the following arithmetico - geometric sequence: 

1, 4x, 7x², 10x³, 13x⁴, …

Find S_n of the following arithmetico - geometric sequence:

1, 2 × 3, 3 × 9, 4 × 27, 5 × 81, …

Find S_n of the following arithmetico - geometric sequence:

3, 12, 36, 96, 240, …

Find the sum to infinity of the following arithmetico - geometric sequence:

`1, 2/4, 3/16, 4/64, ...`

Find the sum to infinity of the following arithmetico - geometric sequence:

`3, 6/5, 9/25, 12/125, 15/625, ...`

Find the sum to infinity of the following arithmetico - geometric sequence:

`1, -4/3, 7/9, -10/27 ...`

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

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

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

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

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

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

Find (70² – 69²) + (68² – 67²) + (66² – 65²) + ... + (2² – 1²)

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

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,` find n

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 + 8 S₁) 

Select the correct answer from the given alternative.

The common ratio for the G.P. 0.12, 0.24, 0.48, is –

Select the correct answer from the given alternative.

The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –

Select the correct answer from the given alternative.

If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?

Select the correct answer from the given alternative.

Which term of the geometric progression 1, 2, 4, 8, ... is 2048

Select the correct answer from the given alternative.

If common ratio of the G.P is 5, 5^th term is 1875, the first term is -

Select the correct answer from the given alternative.

The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is –

Select the correct answer from the given alternative.

Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –

Select the correct answer from the given alternative.

The tenth term of H.P. `2/9, 1/7, 2/19, 1/12, ...` is –

Select the correct answer from the given alternative.

Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)

Select the correct answer from the given alternative.

The G.M.of two numbers exceeds their H.M. by `6/5`, the A.M. exceeds G.M. by `3/2` the two numbers are ...

Answer the following:

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

Answer the following:

Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`

Answer the following:

For a G.P. a = `4/3` and t₇ = `243/1024`, find the value of r

Answer the following:

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.

Answer the following:

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

Answer the following:

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

Answer the following:

For a sequence S_n = 4(7ⁿ – 1) verify that the sequence is a G.P.

Answer the following:

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

Answer the following:

Find the n^th term of the sequence 0.6, 0.66, 0.666, 0.6666, ...

Answer the following:

Find `sum_("r" = 1)^"n" (5"r"^2 + 4"r" - 3)`

Answer the following:

Find `sum_("r" = 1)^"n" "r"("r" - 3)("r" - 2)`

Answer the following:

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

Answer the following:

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

Answer the following:

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

Answer the following:

Find 2 × 5 × 8 + 4 × 7 × 10 + 6 × 9 × 12 + ... upto n terms

Answer the following:

Find `1^2/1 + (1^2 + 2^2)/2 + (1^2 + 2^2 + 3^2)/3 + ...` upto n terms

Answer the following:

Find 12² + 13² + 14² + 15² + ... 20² 

Answer the following:

If `(1 + 2 + 3 + 4 + 5 + ...  "upto n terms")/(1 xx 2 + 2 xx3 + 3 xx 4 + 4 xx5 + ...  "upto n terms") = 3/22` Find the value of n 

Answer the following:

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

Answer the following:

If  `(1 xx 3 + 2 xx 5 + 3 xx 7 + ...  "upto n terms")/(1^3 + 2^3 + 3^3 + ...  "upto n terms") = 5/9`, find the value of n

Answer the following:

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

Answer the following:

If for a G.P. t₃ = `1/3`, t₆ = `1/81` find r

Answer the following:

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

Answer the following:

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

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If for a G.P. first term is (27)² and seventh term is (8)², find S⁸ 

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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

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Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.

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If p, q, r are in G.P. and `"p"^(1/x) = "q"^(1/y) = "r"^(1/z)`, verify whether x, y, z are in A.P. or G.P. or neither.

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If a, b, c are in G.P. and ax² + 2bx + c = 0 and px² + 2qx + r = 0 have common roots then verify that pb² – 2qba + ra² = 0

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If p, q, r, s are in G.P., show that (p² + q² + r²) (q² + r² + s²) = (pq + qr + rs)²   

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If p, q, r, s are in G.P., show that (pⁿ + qⁿ), (qⁿ + rⁿ) , (rⁿ + sⁿ) are also in G.P.

Answer the following:

Find the coefficient of x⁶ in the expansion of e^2x using series expansion

Answer the following:

Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`