Sequence of series practice · Easy

Option D ($\sqrt{z x}>y$) is correct because in a harmonic progression, the harmonic mean is always less than the geometric mean.

Option C (2) is correct because the sum of the series is less than 2, as derived by simplifying the terms and using the properties of series.

Option A ($\frac{\alpha}{\beta}$) is correct because the ratio of the sum of even-indexed terms to the sum of odd-indexed terms in a GP is equal to the common ratio.

Option A ($a=\frac{1}{12}$) is correct because, after expanding the summation and comparing coefficients, the coefficient of $n^4$ is $\frac{1}{12}$.

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.

Option C (1) is correct because when x, y, and z are both in A.P. and G.P., the expression simplifies to 1.

Option C (6603) is correct because the number of digits in $2000^{2000}$ is found by calculating $\log_{10}(2000^{2000})$ and adding 1 to the integer part of the result.

Option C ($x$) is correct because the expression simplifies to x by using the properties of exponents and the sum of an infinite geometric series.

Option A (${\log _a}({{\log }_b}a)$) is correct because it is the solution for $x$ when ${\log _x}a,{a^{x/2}}$ and ${\log _b}x$ are in G.P.

Option C (1/2) is correct because by taking logarithms and manipulating the equations, we can find that 1/x - 1/y equals 1/2.

Option B (2) is correct because the difference between consecutive terms in the A.P. is 2, as derived from the given sum formula.

Option D ($\frac{{4006}}{{3009}}$) is correct because the sum of the series is $\frac{{4006}}{{3009}}$.

Option B (GP) is correct because the solutions to the equation form a geometric progression after simplifying and solving for x.

Option C ($ac + bd > {b^2} + {c^2}$) is correct because in a Harmonic Progression, the relationship $ac + bd > {b^2} + {c^2}$ holds true.

Option B ($q^2=p r$) is correct because in a G.P., if terms are equally spaced, then they are in G.P., which means the square of the middle term equals the product of the other two.

Option C ($\frac{{n + {n^{ - 1}}}}{2}$) is correct because the given series is the expansion of $\frac{{{e^{{{\log }_e}n}} + {e^{ - {{\log }_e}n}}}}{2}$, which simplifies to $\frac{{n + {n^{ - 1}}}}{2}$.

Option C ($\frac{2}{5}$) is correct because the 5th term is calculated using the formula for the nth term of a Geometric Progression (GP).

Option A (2/b) is correct because given the conditions, simplifying the expression 1/x + 1/y results in 2/b.

Option B (-171) is correct because it is the value of lambda calculated by finding the first term and common ratio of the G.P. and then using the sum formula.

Option B (702) is correct because the sum of the two-digit numbers that leave a remainder of 5 when divided by 7 is 702.

Option A ($\log 2$) is correct because simplifying the logarithmic expression using properties of logarithms results in $\log 2$.

Option C (17) is correct because the second term is calculated by subtracting the sum of the first term from the sum of the first two terms.

Option A (2) is correct because when a=b=c=2, abc=8 and a, b, c are in A.P., thus the minimum possible value of b is 2.

Option A (5) is correct because by finding the common difference of the AP and substituting it into the general term, we can identify the number of integral arithmetic means.

Option B is correct because the series can be simplified to an arithmetic progression with a common difference of $\sqrt{2}$, and the sum of the first 24 terms is calculated correctly.

Option A (2) is correct because the given series is the expansion of $e^x$ where x is $log_e 2$.

Option A (3) is correct because the sum to infinity of the given series can be found by manipulating the series and using the formula for the sum of an infinite geometric progression.

Option B ($(14n-6):(8n+23)$) is correct because the ratio of the nth terms is obtained by replacing n by (2n-1) in the given ratio of the sums.

Option A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because the sum to n terms of the given G.P. is correctly calculated using the formula provided in the reason.

Option D ((2490, 248)) is correct because it calculates the sum and the nth term of the arithmetic progression after determining the common difference and the number of terms.