If ${a^2} + 4{b^2} = 12ab,$ then $\log \left( {a + 2b} \right) = $

$\frac{1}{2}\left( {\log a + \log b - 2} \right)$

$\log \frac{a}{2} + \log \frac{b}{2} + \log 2$

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JEE • MATHEMATICS • SEQUENCE OF SERIES

Sequence of series practice · Mixed difficulty

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 301-330 of 998 questions

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301 Easy

301 of 998

Application +4 | -1

\frac{{5 + 9 + 13 + ... + n\;\mathrm{terms}}}{{7 + 9 + 11 + ... + 12\;\mathrm{terms}}} = \frac{5}{{12}}

, then

n

is equal to

View answer and explanation

Correct option: B

Option B (6) is correct because solving the equation formed by equating the ratio of the sums of the two arithmetic progressions to 5/12 yields n=6.

302 Medium

302 of 998

Application +4 | -1

The sum of

0.2+0.004+0.00006+0.0000008+\ldots i s

View answer and explanation

Correct option: B

\frac{2000}{9801}

Option B (

\frac{2000}{9801}

) is correct because the sum of the given series, which is an arithmetico-geometric progression, is calculated correctly using the appropriate formula.

303 Easy

303 of 998

Application +4 | -1

\log _{10} 2, \log _{10}\left(2^x-1\right)

and

\log _{10}\left(2^x+3\right)

are three consecutive terms of an A.P. for

View answer and explanation

Correct option: C

exactly one real

x

Option C (exactly one real

x

) is correct because the solution to the equation derived from the arithmetic progression property yields only one real value for x.

304 Medium

304 of 998

Application +4 | -1

If 19th term of a non-zero AP is zero, then its (49th term) : (29th term) is

View answer and explanation

Correct option: D

03:01

Option D (03:01) is correct because using the given condition that the 19th term is zero, the ratio of the 49th term to the 29th term is 3:1.

305 Hard

305 of 998

Understanding +4 | -1

S = \sum \limits_{n = 0}^\infty \frac{{{{\left( {\log x} \right)}^{2n}}}}{{\left( {2\;n} \right)\;!}},

then

S

equals

View answer and explanation

Correct option: C

\frac{1}{2}\left( {x + {x^{ - 1}}} \right)

Option C is correct because the given series is the Taylor series expansion of cosh(log x), which simplifies to (x + x^(-1))/2.

306 Medium

306 of 998

Analyzing +4 | -1

\left| a \right| < 1

and

\left| b \right| < 1,

then the sum of the series

a\left( {a + b} \right) + {a^2}\left( {{a^2} + {b^2}} \right) + {a^3}\left( {{a^3} + {b^3}} \right) + \ldots

upto

\infty

, is

View answer and explanation

Correct option: B

\frac{{{a^2}}}{{1 - {a^2}}} + \frac{{ab}}{{1 - ab}}

Option B (

\frac{{{a^2}}}{{1 - {a^2}}} + \frac{{ab}}{{1 - ab}}

) is correct because the series can be split into two geometric series, and their sums can be calculated.

307 Medium

307 of 998

Analyzing +4 | -1

x,y,z

are in G.P. and

x + 3,y + 3,z + 3

are in H.P., then

View answer and explanation

Correct option: B

y = 3

Option B (

y = 3

) is correct because substituting

y=3

into the given conditions satisfies both the geometric and harmonic progression relationships.

308 Easy

308 of 998

Application +4 | -1

{\log _3}x \times {\log _x}2x \times {\log _{2x}}y = {\log _x}{x^2},

then

y

equals

View answer and explanation

Correct option: A

Option A (9) is correct because, after simplifying the logarithmic expression using the change of base formula, we find that

y = 3^2 = 9

309 Medium

309 of 998

Analyzing +4 | -1

a,b,c

are three unequal positive quantities in H.P., then

View answer and explanation

Correct option: B

{a^5} + {c^5} > 2{b^5}

Option B (

{a^5} + {c^5} > 2{b^5}

) is correct. Since

a, b, c

are in H.P.,

{a^n} + {c^n} > 2{b^n}

holds for

n > 1

, and thus

{a^5} + {c^5} > 2{b^5}

is true.

310 Easy

310 of 998

Application +4 | -1

{a^2} + 4{b^2} = 12ab,

then

\log \left( {a + 2b} \right) =

View answer and explanation

Correct option: C

\frac{1}{2}\left( {\log a + \log b + 4\log 2} \right)

Option C (

\frac{1}{2}(\log a + \log b + 4\log 2)

) is correct because it is the result of applying logarithmic properties to the given equation.

311 Medium

311 of 998

Analyzing +4 | -1

The sum of the infinite series

1 + \frac{1}{{2!}} + \frac{{1.3}}{{4!}} + \frac{{1.3.5}}{{6!}} + \ldots

View answer and explanation

Correct option: C

\sqrt e

Option C (

\sqrt e

) is correct. The given series can be rewritten as the expansion of

e^{1/2}

, which simplifies to

\sqrt{e}

312 Easy

312 of 998

Understanding +4 | -1

I. If each term of a G.P. be raised to the same power, the resulting sequence also forms a G.P.

II.

25^{\text {th}}

term of the sequence 4, 9, 14, 19,.... is 124.

Choose the correct option.

View answer and explanation

Correct option: A

Both are true

Option A (Both are true) is correct because both statements are fundamental properties of geometric progressions and arithmetic progressions.

313 Medium

313 of 998

Application +4 | -1

If the interior angles of a polygon are in A.P. with common difference

5^\circ

and the smallest angle

120^\circ

, then the number of sides of the polygon is

View answer and explanation

Correct option: B

\mathrm{9}

Option B (9) is correct because the number of sides of the polygon is 9, considering the constraint that the interior angles must be less than 180 degrees.

314 Easy

314 of 998

Understanding +4 | -1

The largest interval for which the series

1+(x-1)+(x-1)^{2}

+\cdots \infty

may be summed is

View answer and explanation

Correct option: B

0 < x < 2

Option B (

0 < x < 2

) is correct because the series converges when the absolute value of the common ratio (x-1) is less than 1.

315 Easy

315 of 998

Application +4 | -1

Three numbers are in A.P, such that their sum is 18 and sum of their squares is 158. The greatest among them is

View answer and explanation

Correct option: B

Option B (11) is correct because it is the largest of the three numbers in arithmetic progression that satisfy the given conditions.

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316 Medium

316 of 998

Application +4 | -1

If 10 th term of a GP is 9 and 4 th term is 4 , then its 7 th term is

View answer and explanation

Correct option: A

Option A (6) is correct because the 7th term of the GP can be found by using the given 10th and 4th terms.

317 Hard

317 of 998

Analyzing +4 | -1

\frac{1}{{{e^{3x}}}}\left( {{e^x} + {e^{5x}}} \right) = {a_0} + {a_1}x + {a_2}{x^2} + \ldots

, then

2{a_1} + {2^3}{a_3} + {2^5}{a_5} + \ldots

is equal to

View answer and explanation

Correct option: D

Option D (0) is correct because when the given expression is simplified, the coefficients of the odd powers of x are zero, resulting in the sum being zero.

318 Medium

318 of 998

Application +4 | -1

If the sum of four integers in AP is 24 and their product is 945 , then the numbers are

View answer and explanation

Correct option: A

3,5,7,9

Option A (

3,5,7,9

) is correct because the sum of these integers in AP is 24 and their product is 945.

319 Easy

319 of 998

Application +4 | -1

The sum of the series

{1^3} + {2^3} + {3^3} + ... + {15^3}

View answer and explanation

Correct option: C

14400

Option C (14400) is correct because the sum of the cubes of the first 15 natural numbers is 14400.

320 Easy

320 of 998

Application +4 | -1

The value of

\frac{1}{{2!}} + \frac{2}{{3!}} + ... + \frac{{999}}{{1000!}}

is equal to

View answer and explanation

Correct option: A

\frac{{1000! - 1}}{{1000!}}

Option A (

\frac{{1000! - 1}}{{1000!}}

) is correct because the series telescopes, leaving only the first term and the negative of the last term.

321 Medium

321 of 998

Application +4 | -1

Sum to n terms of the series

\frac{1}{{1\;.\;\;2\;.\;\;3\;.\;\;4}} + \frac{1}{{2\;.\;\;3\;.\;\;4\;.\;\;5\;}} + \frac{1}{{3\;.\;\;4\;.\;\;5\;.\;6\;}} + \ldots ,

View answer and explanation

Correct option: D

\frac{{{n^3} + 6{n^2} + 11n}}{{18\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right)}}

Option D (

\frac{{{n^3} + 6{n^2} + 11n}}{{18\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right)}}

) is correct because it represents the sum to n terms of the given series after using partial fractions.

322 Easy

322 of 998

Application +4 | -1

The sum of the infinite series

\frac{5}{17}+\frac{55}{(17)^2}+\frac{555}{(17)^3}+\ldots

View answer and explanation

Correct option: B

\frac{85}{112}

Option B (85/112) is correct because it is obtained by manipulating the infinite series and using the formula for the sum of an infinite geometric progression.

323 Medium

323 of 998

Analyzing +4 | -1

k \ne 0

is a constant and

n \in N

, then,

\lim\limits_{n \to \infty } \,\,\left\{ {\frac{{n!}}{{{{\left( {k\;n} \right)}^n}}}} \right\}

is equal to

View answer and explanation

Correct option: D

\frac{1}{{ke}}

Option D (

\frac{1}{{ke}}

) is correct because the limit is evaluated using Stirling's approximation and properties of logarithms and integrals.

324 Medium

324 of 998

Application +4 | -1

Let

\mathrm{G}

be the geometric mean of two positive numbers a and

\mathrm{b}

, and

\mathrm{M}

be the arithmetic mean of

\frac{1}{\mathrm{a}}

and

\frac{1}{\mathrm{b}}

. If

\frac{1}{\mathrm{M}}: \mathrm{G}

4: 5

, then

a: b

can be:

View answer and explanation

Correct option: A

1: 4

Option A (1:4) is correct because it satisfies the given condition

\frac{1}{M}: G = 4:5

after simplifying the geometric and arithmetic means.

325 Medium

325 of 998

Analyzing +4 | -1

The sum of

\frac{{\frac{1}{2} \cdot \frac{2}{2}}}{{{1^3}}} + \frac{{\frac{2}{2}.\frac{3}{2}}}{{{1^3} + {2^3}}} + \frac{{\frac{3}{2}.\frac{4}{2}}}{{{1^3} + {2^3} + {3^3}}} + ...

upto

n

terms is equal to

View answer and explanation

Correct option: B

\frac{n}{{n + 1}}

Option B (

\frac{n}{{n + 1}}

) is correct because the series telescopes to this result after simplification and partial fraction decomposition.

326 Easy

326 of 998

Application +4 | -1

The sum to

n

terms of the series

\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\ldots

, is

View answer and explanation

Correct option: C

\frac{n}{2 n+1}

Option C (

\frac{{n}}{{2n+1}}

) is correct because the series can be expressed as a telescoping series, where most terms cancel out, leaving a simple expression in terms of

n

327 Medium

327 of 998

Application +4 | -1

x, y

and

z

are in HP, where

z>y>x

. Then, value of

\log (x+z)+\log (x-2 y+z)

View answer and explanation

Correct option: B

2 \log (z-x)

Option B (

2 \log (z-x)

) is correct because using the properties of Harmonic Progression, we can simplify the expression to obtain the result.

328 Easy

328 of 998

Understanding +4 | -1

x, y, z

are in harmonic progression, then which of the following is true?

View answer and explanation

Correct option: D

\sqrt{z x}>y

Option D (

\sqrt{z x}>y

) is correct because in a harmonic progression, the harmonic mean is always less than the geometric mean.

329 Hard

329 of 998

Application +4 | -1

The coefficient of

{x^n}

in the expansion of

\frac{{{e^{7\;x}} + {e^x}}}{{{e^{3\;x}}}},

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the coefficient of x^n in the expansion is

\frac{{{{4^n}} + {{\left( { - 2} \right)}^n}}}{{n\\;!}}

, which doesn't match any of the given options.

330 Easy

330 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: C

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

Option C is correct because the sum of the given series is derived by converting the decimals to fractions and using the formula for the sum of a geometric progression.

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