If $p,q,r$ are in A.P., then $p{\rm{th}},q{\rm{th}}$ and $r{\rm{th}}$ terms of any G.P. are in

Reciprocals of these terms are in A.P.

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JEE • MATHEMATICS • SEQUENCE OF SERIES

Sequence of series practice · Easy

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Showing 1-30 of 998 questions

1 Easy

1 of 998

Analyzing +4 | -1

a,b,c

are in A.P.;

a,x,b

are in G.P. and

b,y,c

are in G.P., then

{x^2},{b^2},{y^2}

are in

View answer and explanation

Correct option: C

A.P.

Option C (A.P.) is correct because, given the conditions, it can be shown that

{x^2},{b^2},{y^2}

satisfy the arithmetic progression relationship.

2 Easy

2 of 998

Application +4 | -1

Suppose

P(x)=1-x+x^2-x^3+\ldots+x^{2012}

is expressed as a polynomial in

y

, as

Q(y)=a_0+a_1 y+\ldots+a_{2012} y^{2012}

where

y=x-2

, then

\sum\limits_{{i=0}}^{2012} a_i

equals

View answer and explanation

Correct option: A

\frac{1}{4}\left(3^{2013}+1\right)

Option A is correct because the sum of the coefficients of the polynomial Q(y) is equal to P(3), which can be calculated using the formula for P(x).

3 Easy

3 of 998

Application +4 | -1

{x^{2{{\log }_{10}}x}} = 1000x,

, then

x

equals to

View answer and explanation

Correct option: B

{10^{ - 1}},10\sqrt {10}

Option B (

{10^{ - 1}},10\sqrt {10}

) is correct because these are the two values of x that satisfy the given equation

{x^{2{{\log }_{10}}x}} = 1000x

4 Easy

4 of 998

Application +4 | -1

p,q,r

are in A.P., then

p\mathrm{th},q\mathrm{th}

and

r\mathrm{th}

terms of any G.P. are in

View answer and explanation

Correct option: B

G.P.

Option B (G.P.) is correct because if p, q, r are in A.P., then the pth, qth, and rth terms of any G.P. will also be in G.P.

5 Easy

5 of 998

Application +4 | -1

If the

p^{\text {th }}

term of an AP is

q

and the

q^{\text {th }}

term is

p

, then its

r^{\text {th }}

term will be

View answer and explanation

Correct option: B

p+q-r

Option B (

p+q-r

) is correct because by finding the common difference and the first term of the AP, the

r^{th}

term can be expressed as

p+q-r

6 Easy

6 of 998

Application +4 | -1

The ratio of sum of first three terms of a G.P.to the sum of first six terms is

64: 91

, the common ratio of G.P. is

View answer and explanation

Correct option: B

\frac{3}{4}

Option B (

\frac{{3}}{{4}}

) is correct because it is the common ratio obtained by setting up the ratio of the sums of the first three and six terms of the G.P. and solving for r.

7 Easy

7 of 998

Application +4 | -1

\log \left( {2a - 3b} \right) = \log a - \log b,

then

a =

View answer and explanation

Correct option: A

\frac{{3{b^2}}}{{2b - 1}}

Option A is correct because by applying the properties of logarithms and simplifying the equation, we can isolate 'a' and express it in terms of 'b'.

8 Easy

8 of 998

Application +4 | -1

Sum of

n

terms of the series

\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{{15}}{{16}} + ...

View answer and explanation

Correct option: D

{2^{ - n}} + n - 1

Option D (

{2^{ - n}} + n - 1

) is correct because it is derived by summing the given series after rewriting each term.

9 Easy

9 of 998

Application +4 | -1

a,\;b,\;c

are in AP then

{10^{ax + 10}},\;{10^{bx + 10}},\;{10^{cx + 10}}\left( {x \ne 0} \right)

are in

View answer and explanation

Correct option: C

GP for all

x

Option C (GP for all x) is correct because the given terms form a GP regardless of the value of x, as shown by the relationship derived from the AP condition.

10 Easy

10 of 998

Application +4 | -1

The 10th common term between the series

3+7+11+\ldots

and

1+6+11+\ldots .

, is

View answer and explanation

Correct option: A

191

Option A (191) is correct because it finds the 10th common term of the two given arithmetic series.

11 Easy

11 of 998

Analyzing +4 | -1

x > 1,\;y > 1,\;z > 1

are in GP, then

\frac{1}{{1 + \log x}},\frac{1}{{1 + \log y}},\frac{1}{{1 + \log z}}

are in

View answer and explanation

Correct option: B

Option B (HP) is correct because if x, y, z are in GP, then

\frac{1}{{1 + \log x}},\frac{1}{{1 + \log y}},\frac{1}{{1 + \log z}}

are in Harmonic Progression.

12 Easy

12 of 998

Application +4 | -1

If the angles of a quadrilateral are in AP, whose common difference is

10^\circ

, then the angles of the quadrilateral are

View answer and explanation

Correct option: B

75^\circ ,85^\circ ,95^\circ ,105^\circ

Option B (

75^\circ ,85^\circ ,95^\circ ,105^\circ

) is correct because these angles are in AP with a common difference of

10^\circ

and sum up to

360^\circ

13 Easy

13 of 998

Application +4 | -1

If the sum of first

n

natural numbers is 1/5 times the sum of their squares, then the value of

n

View answer and explanation

Correct option: C

Option C (7) is correct because solving the equation derived from the given relationship between the sum of first n natural numbers and the sum of their squares yields n=7.

14 Easy

14 of 998

Application +4 | -1

If three positive real numbers

a, b, c

are in

A P

such that

a b c=4

, then the minimum value of

b

View answer and explanation

Correct option: B

2^{2 / 3}

Option B (

2^{2 / 3}

) is correct because it is the minimum value of b, obtained by using the arithmetic progression property and minimizing the expression abc=4.

15 Easy

15 of 998

Analyzing +4 | -1

If the expression

\left.\exp \left\{1+|\cos x|+\cos ^2 x+\left|\cos ^3 x\right|+\cos ^4 x+\ldots \infty\right) \log _e 4\right\}

satisfies the equation

y^2-20 y+64=0

for

0 < x < \pi

, then the set of values of

x

View answer and explanation

Correct option: D

\{\pi / 3, \pi / 2,2 \pi / 3\}

Option D (

\{\pi / 3, \pi / 2,2 \pi / 3\}

) is correct because these are the values of x in the given range that satisfy the equation derived from the exponential expression and the quadratic equation.

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16 Easy

16 of 998

Application +4 | -1

If the sides of a right-angled triangle are in AP, then the sines of the acute angles are

View answer and explanation

Correct option: A

3 / 5,4 / 5

Option A (

3 / 5,4 / 5

) is correct because by assuming the sides are in A.P. and using the Pythagorean theorem, the sines of the acute angles can be determined as 3/5 and 4/5.

17 Easy

17 of 998

Application +4 | -1

In an arithmetic progression of 16 distinct terms with

a_1=16

, the sum is equal to square of the last term. The common difference of the A.P. is

View answer and explanation

Correct option: C

-8 / 5

Option C (-8/5) is correct because, given the conditions, the common difference of the A.P. is -8/5.

18 Easy

18 of 998

Analyzing +4 | -1

{2^{1/4}} \cdot {4^{1/16}}{8^{1/48}}.....

\infty

is equal to

View answer and explanation

Correct option: A

\sqrt{2}

Option A (

\sqrt{2}

) is the correct answer because the expression simplifies to

2^{1/2}

, which is the square root of 2.

19 Easy

19 of 998

Application +4 | -1

If AM and GM of

x

and

y

are in the ratio

p:q

, then

x:y

View answer and explanation

Correct option: B

p + \sqrt {{p^2} - {q^2}} :p - \sqrt {{p^2} - {q^2}}

Option B (

p + \sqrt {{p^2} - {q^2}} :p - \sqrt {{p^2} - {q^2}}

) is correct because it's derived from the given ratio of AM and GM using componendo and dividendo.

20 Easy

20 of 998

Application +4 | -1

a_1, a_2, a_3, \ldots

are in A.P. If

\frac{a_1+a_2+a_3+\ldots+a_p}{a_1+a_2+a_3+\ldots+a_q}=\frac{p^2}{q^2}

, where

p \neq q

, then

\frac{a_6}{a_{21}}=

View answer and explanation

Correct option: B

\frac{11}{41}

Option B (

\frac{{11}}{{41}}

) is correct because by using the formula for the sum of an arithmetic progression and comparing ratios, we can find the ratio of the 6th term to the 21st term.

21 Easy

21 of 998

Analyzing +4 | -1

x = {\log _3}5,\;y = {\log _{17}}25

which one of the following is correct?

View answer and explanation

Correct option: C

x > y

Option C (

x > y

) is correct because by converting the logarithms to a common base and comparing, it can be shown that x is greater than y.

22 Easy

22 of 998

Analyzing +4 | -1

Let

a_n

be the

n^{\text {th }}

term of the G.P. of positive numbers. Let

\sum\limits_{n=1}^{100} a_{2 n}=\alpha

and

\sum\limits_{n=1}^{100} a_{2 n-1}=\beta

, such that

\alpha \neq \beta

, then the common ratio is

View answer and explanation

Correct option: A

\alpha / \beta

Option A (

\alpha / \beta

) is correct because the common ratio of the G.P. can be found by dividing the sum of even-indexed terms by the sum of odd-indexed terms.

23 Easy

23 of 998

Application +4 | -1

Sum to 25 terms of the series

0.5 + 0.55+ 0.555 + \ldots

is:

View answer and explanation

Correct option: A

\frac{5}{{81}}\left( {224 - {{10}^{ - 25}}} \right)

Option A (

\frac{5}{{81}}\left( {224 - {{10}^{ - 25}}} \right)

) is correct because by converting the decimal series into a geometric progression, we can find the sum to 25 terms.

24 Easy

24 of 998

Analyzing +4 | -1

a, x, b

are in A.P.,

a, y

b

are in G.P. and

a, z, b

are in H.P. such that

x=9 z

and

a > 0, b > 0

, then

View answer and explanation

Correct option: A

|y|=3 z

and

x=3|y|

Option A (

|y| = 3z

and

x = 3|y|

) is correct because it follows from the relationships between arithmetic, geometric, and harmonic means.

25 Easy

25 of 998

Application +4 | -1

a, b, c

be in A.P. and

b, c, d

be in H.P., then

View answer and explanation

Correct option: B

a d=b c

Option B (

a d=b c

) is correct because using the properties of arithmetic and harmonic progressions, we can derive the relationship

ad = bc

26 Easy

26 of 998

Application +4 | -1

If the sum of the series

1 + \frac{3}{x} + \frac{9}{{{x^2}}} + \frac{{27}}{{{x^3}}} + \ldots \mathrm{to\;}\infty

is a finite number, then

View answer and explanation

Correct option: D

x > 3

Option D (

x > 3

) is correct because the series converges only when the absolute value of the common ratio is less than 1, which implies

|\frac{3}{x}| < 1

27 Easy

27 of 998

Analyzing +4 | -1

\left|\begin{array}{ccc}a & b & a \alpha-b \\ b & c & b \alpha-c \\ 2 & 1 & 0\end{array}\right|=0

and

\alpha \neq 1 / 2

, then

a, b, c

are in

View answer and explanation

Correct option: B

G.P.

Option B (G.P.) is correct because the determinant condition leads to the relationship

ac = b^2

, which is the defining property of a geometric progression.

28 Easy

28 of 998

Application +4 | -1

Value of

y=(0.36)^{\log _{0.25}\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\ldots \text { upto } \infty\right)}

View answer and explanation

Correct option: C

0.6

Option C (0.6) is correct because after simplifying the geometric series and applying logarithmic properties, the value of y is found to be 0.6.

29 Easy

29 of 998

Application +4 | -1

The sum of first 20 terms of the sequence

0.7,0.77,0.777, \ldots

, is

View answer and explanation

Correct option: B

\frac{7}{81}\left(179+10^{-20}\right)

Option D (

\frac{7}{81}(179+10^{-20})

) is correct because it represents the sum of the first 20 terms of the given sequence, calculated by converting the terms into a form suitable for geometric progression.

30 Easy

30 of 998

Analyzing +4 | -1

{\log _a}x,{\log _b}x,{\log _c}x\;

be in HP, then

a,\;b,\;c

are in

View answer and explanation

Correct option: C

Option C (GP) is correct because if

{\log _a}x,{\log _b}x,{\log _c}x

are in HP, then

a, b, c

are in GP.

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Key Takeaways

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