Sequences and Series - Maharashtra Board STD 11 Science Maths Questions

Q 1MCQ1 Mark

The G.M. of two numbers exceeds their H.M. by $\frac{6}{5}$, the A.M. exceeds G.M. by $\frac{3}{2}$ the two numbers are

A
$6, \frac{15}{2}$
B
15, 25
✓
3, 12
D
$\frac{6}{5}, \frac{3}{2}$

Answer: C.

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Q 2MCQ1 Mark

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

A
$A =\frac{a+b}{2}$
B
$G=\sqrt{a b}$
C
$H =\frac{2 a b}{a+b}$
✓
A = GH

Answer: D.

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Q 3MCQ1 Mark

The tenth term of H.P. $\frac{2}{9}, \frac{1}{7}, \frac{2}{19}, \frac{1}{12}, \ldots$ is

✓
$\frac{1}{27}$
B
$\frac{9}{2}$
C
$\frac{5}{2}$
D
27

Answer: A.

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Q 4MCQ1 Mark

Sum to infinity of a G.P. $5,-\frac{5}{2}, \frac{5}{4},-\frac{5}{8}, \frac{5}{16}, \ldots$ is

A
5
B
$-\frac{1}{2}$
✓
$\frac{10}{3}$
D
$\frac{3}{10}$

Answer: C.

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Q 5MCQ1 Mark

The sum of 3 terms of a G.P. is $\frac{21}{4}$ and their product is 1 , then the common ratio is

A
1
B
2
✓
4
D
8

Answer: C.

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Q 6Solve the following Question.(1 Marks)1 Mark

Find the coefficient $x^6$ in the expression of $e^{2 x}$ using series expansion.

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Q 7Solve the following Question.(1 Marks)1 Mark

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

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Q 8Solve the following Question.(1 Marks)1 Mark

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

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Q 9Solve the following Question.(1 Marks)1 Mark

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

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Q 10Solve the following Question.(1 Marks)1 Mark

For a G.P. $a=\frac{4}{3}$ and $t_7=\frac{243}{1024}$, find the value of $r$.

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Q 11Solve the Following Question.(2 Marks)2 Marks

If $p, q, r, s$ are in G.P., show that $\left(p^2+q^2+r^2\right)\left(q^2+r^2+s^2\right)=(p q+q r+r s)^2$.

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Q 12Solve the Following Question.(2 Marks)2 Marks

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

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Q 13Solve the Following Question.(2 Marks)2 Marks

If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of

$x^{q-r} \cdot y^{r-p} \cdot z^{p-q}$

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Q 14Solve the Following Question.(2 Marks)2 Marks

If for a G.P. first term is $(27)^2$ and the seventh term is $(8)^2$, find $S_8$.

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Q 15Solve the Following Question.(2 Marks)2 Marks

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

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Q 16Solve the Following Question.(3 Marks)3 Marks

If $p, q, r, s$ are in G.P., show that $\left(p^n+q^n\right),\left(q^n+r^n\right),\left(r^n+s^n\right)$ are also in G.P.

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Q 17Solve the Following Question.(3 Marks)3 Marks

If $a, b, c$ are in G.P. and $a x^2+2 b x+c=0$ and $p x^2+2 q x+r=0$ have common roots, then

verify that $p^2-2 q b a+r a^2=0$.

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Q 18Solve the Following Question.(3 Marks)3 Marks

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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Q 19Solve the Following Question.(3 Marks)3 Marks

Find $\left(50^2-49^2\right)+\left(48^2-47^2\right)+\left(46^2-45^2\right)+\ldots+\left(2^2-1^2\right)$.

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Q 20Solve the Following Question.(3 Marks)3 Marks

If $\frac{1+2+3+4+5+\ldots \text { upto } n \text { terms }}{1 \times 2+2 \times 3+3 \times 4+4 \times 5+\ldots \text { upto } n \text { terms }}=\frac{3}{22}$, Find the value of $n$.

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Q 21Solve the Following Question.(4 Marks)4 Marks

Find the sum of infinite terms of $1+\frac{4}{5}+\frac{7}{25}+\frac{10}{125}+\frac{13}{625}+\ldots$

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Q 22Solve the Following Question.(4 Marks)4 Marks

Find Sn of the following arithmetico-geometric sequences. : 3, 12, 36, 96, 240, ……

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Q 23Solve the Following Question.(4 Marks)4 Marks

Find Sn of the following arithmetico-geometric sequences. : 1, 2 × 3, 3 × 9, 4 × 27, 5 × 81, ……

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Q 24Solve the Following Question.(4 Marks)4 Marks

Find Sn of the following arithmetico-geometric sequences..

$1,4 x, 7 x^2, 10 x^3, 13 x^4$

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Q 25Solve the Following Question.(4 Marks)4 Marks

Find Sn of the following arithmetico-geometric sequences.

$2,4 x, 6 x^2, 8 x^3, 10 x^4, \ldots \ldots$

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Q 26Solve the Following Question.(5 Marks)5 Marks

If $\frac{1 \times 3+2 \times 5+3 \times 7+\ldots \text { upto } n \text { terms }}{1^3+2^3+3^3+\ldots \text { upto } n \text { terms }}=\frac{5}{9}$, find the value of $n$.

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Q 27Solve the Following Question.(5 Marks)5 Marks

Find two numbers whose A.M. exceeds their G.M. by $\frac{1}{2}$ and their H.M. by $\frac{25}{26}$.

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Q 28Solve the Following Question.(5 Marks)5 Marks

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

1. the areas of all the squares.

2. the perimeters of all the squares.

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Q 29Solve the Following Question.(5 Marks)5 Marks

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

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