Sequence of series practice · Easy

Option C ($\frac{{3n - 2}}{{{{5^{n - 1}}}}}$) is correct because the nth term of the series is derived from the corresponding Arithmetic Progression (AP) and Geometric Progression (GP).

Option A ($\frac{{13}}{4}$) is correct because by using the method of difference to solve the infinite series, we can find the sum to be $\frac{{13}}{4}$.

Option C (27) is correct because solving the equation $T_m = S_m - S_{m-1} = 164$ yields $m = 27$.

Option C ($\frac{1}{2}\left( {\log a + \log b + 4\log 2} \right)$) is correct because by completing the square and applying logarithm properties, we can derive the given expression.

Option B (1/3) is correct because solving the equation derived from the A.P. condition yields r=1/3.

Option D (None of these) is correct because the only value of $\alpha$ that satisfies the given condition, $\alpha = \frac{{\pi}}{{2}}$, does not lie in any of the provided intervals.

Option B ($\frac{(n-1) r}{1-r^2}$) is correct because it represents the sum of the given series where the terms are in GP, derived by simplifying the general term and summing.

Option A ($\frac{{{\pi ^4}}}{{96}}$) is correct because it is derived by subtracting the sum of even terms from the total sum of the series.

Option B (45/13, 45/11) is correct because these are the two harmonic means between 3 and 5, calculated using the formula for inserting harmonic means.

Option D ($\frac{{(1001) !}}{{(50) !(951) !}}$) is correct because by simplifying the given series and finding the coefficient of $x^{50}$ in the resulting binomial expansion, we get $\frac{{(1001) !}}{{(50) !(951) !}}$.

Option C (3) is correct because the logarithmic expression simplifies to ${\log _5}(125)$, which equals 3.

Option C (H.P.) is correct because the reciprocals of the logarithms form an arithmetic progression, indicating that the original logarithms are in harmonic progression.

Option C (${e^a}\frac{{{b^n}}}{{n\;!}}$) is correct because the coefficient of ${x^n}$ in the given series is found by expanding the exponential function.

Option D (40) is correct because applying AM >= GM inequality yields the least value of 3x + 2y as 40.

Option C (1330) is correct because after calculating the value of n using the geometric mean formula, the summation simplifies to 1330.

Option D (19680) is correct because the sum of the first eight terms of the GP is 19680.

Option B is correct because the given expression can be simplified using the definition of H_n.

Option C (2-$\frac{11}{2^{19}}$) is correct because it is the result of summing the given series using the method of differences.

Option A ($\frac{{1}}{{3}}(\sqrt{{7+3n}}-\sqrt{{7}})$) is correct because rationalizing the denominator of each term and summing the telescoping series leads to this result.

Option C ($\frac{n}{2}\log \left( {\frac{{{a^{n + 1}}}}{{{b^{n - 1}}}}} \right)$) is correct because by applying logarithm properties and simplifying the summation, the expression is equivalent to the given option.

Option B ($x^2-18 x+16=0$) is correct because, given the arithmetic and geometric means, the quadratic equation is formed using the formula $x^2 - 2Ax + G^2 = 0$.

Option C (5) is correct because solving the summation and substituting into the given equation yields n=5.

Option B (120/121) is correct because it is the sum of the 10 terms of the given series, calculated by expressing each term as a difference of two terms and then summing them.

Option C (HP) is correct because the reciprocals of the terms form an arithmetic progression, satisfying the condition for a harmonic progression.

Option A ($\frac{n(n+1)}{2}$) is correct because the sum of the series can be simplified to this expression when n is odd.

Option B ($1 < r < \sqrt{2}$) is correct because the sine rule and the given angle relationship lead to this inequality for the common ratio r.

Option D (1/4 - 1/(2√2)) is correct because it is the value of 'a' derived by using the properties of A.P. and G.P. and solving the resulting quadratic equation.

Option A ($7^{-3}$) is correct because solving the logarithmic equation leads to two possible values for x, and $7^{-3}$ is the least value.

Option C ($\frac{{n\left( {n + 3} \right)}}{2}$) is correct because it is derived by finding the general term ${a_r}$ and then summing the square root of the general term.

Option C is correct because the reciprocals of terms in a Harmonic Progression (H.P.) form an Arithmetic Progression (A.P.), and manipulating the given expression using this property leads to the conclusion that the given terms are in H.P.