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

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, find 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 t₇.

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

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

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, `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 the G. P. is y and eleventh term of the 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.

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 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., 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 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 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 (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" = "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.

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 decimal 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.

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`.

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 × 7 + 9 × 11 + 13 × 15 + ... upto n terms.

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

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

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 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).

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(7ⁿ – 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 12² + 13² + 14² + 15² + … + 20².

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

In a G.P., if t₂ = 7, t₄ = 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}.