Sequence of series practice · Mixed difficulty

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 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 ($q^2=p r$) 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 ($\frac{n+1}{2}$) is correct because the nth term is the sum of the first n integers divided by n, which simplifies to (n+1)/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 C (52) is correct because by using the properties of arithmetic progressions and the given conditions, we can find the common difference and then calculate a_10.

Option A (1) is correct because the expression ${a^3} + {b^3} + {c^3} - 3abc$ can be factored, and using properties of complex numbers and exponential functions, it simplifies to 1.

Option B ($\frac{4}{9}\left( {{{10}^{2n}} - 1} \right)$) is correct because, by converting the repeating digits into fractions and simplifying, the expression equals $\frac{4}{9}\left( {{{10}^{2n}} - 1} \right)$.

Option B ($G P$) is correct because taking the reciprocal of each term of a geometric progression results in another geometric progression.

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

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.

Option D ($\frac{n}{a_1 a_{n+1}}$) is correct because it simplifies the given expression using the properties of arithmetic progressions.

Option C (0,10) is correct because the first term of an infinite G.P. whose sum is five must lie between 0 and 10, derived from the condition -1 < r < 1.

Option A (310) is correct because it is the maximum sum of the series, obtained by summing only the non-negative terms of the AP.

Option A ($\frac{{2355}}{{999}}$) is correct because converting the repeating decimal to a fraction and adding it to the integer part yields $\frac{{2355}}{{999}}$.

Option B (4942) is correct because the 96th term is calculated using the formula for the nth term of the given sequence.

The correct answer is ${x^2} + x - 1 = 0$ because by using logarithm properties, the equation can be simplified to this quadratic form.

Option A ($2+\sqrt{3}: 2-\sqrt{3}$) is correct because it is derived from the relationship between arithmetic and geometric means given the condition A = 2G.