Sequence of series practice · Easy

Option D ($\frac{1}{{72}}$) is correct because the sum of the infinite series is found by expressing the general term as a difference of two terms and then summing the series.

Option B ($\frac{{n-(n+1) x+x^{{n+1}}}}{{(1-x)^2}}$) is correct because it is the result of summing the given series using the formula for a geometric series.

Option B ((4,9)) is correct because the geometric mean is 6 and the arithmetic mean is 6.5, satisfying both conditions.

Option D (a composite number) is correct because the expression for Tr simplifies to a product of two integers, indicating a composite number.

Option C (16) is correct because it is the minimum value of x+y derived from the AM-GM inequality and the given logarithmic condition.

Option C ($\frac{{10 a+1}}{{10+a}}$) is correct because it represents the ratio of the first and last harmonic means in the given series.

Option A ($c^3 a=b^3 d$) is correct because it is derived from the relationship between the roots and coefficients of the cubic equation when the roots are in G.P.

Option A (255) is correct because the product expands to a sum of powers of 'a' up to $a^{255}$, making n = 255.

Option C ($\frac{{p^2-m^2}}{{2 S-(m+p)}}$) is correct because it expresses the common difference in terms of the first term (m), last term (p), and the sum of the first n terms (S).

Option A ($\left( {\frac{{{x^{20}} - 1}}{{{x^2} - 1}}} \right)\left( {\frac{{{x^{22}} + 1}}{{{x^{20}}}}} \right) + 20$) is correct because it represents the sum of the first 10 terms of the given series after simplification.

Option B (7) is correct because solving the given equations for the sum of extreme terms and the product of middle terms yields the greatest number as 7.

Option D (None of these) is correct because solving the logarithmic equation yields $x = 0$, which is not among the provided options.

Option B ($\left( {n - 1} \right){x_1}{x_n}$) is correct because the sum of the product of consecutive terms in a harmonic progression is equal to (n-1) times the product of the first and last terms.

Option B ($4^5$) is correct because the product of the first five terms of a G.P. with the third term being 4 is $4^5$.

Option B is correct because when x=0, the given equation simplifies to e^(e0) = e¹ = e, which equals a_0.

Option C ($\frac{{{a_1}}}{{{a_2}}}$) is correct because by expressing each term of the numerator and denominator in terms of $a_1$ and the common ratio r, the expression simplifies to $\frac{a_1}{a_2}$.

Option G.P. is correct because if $b-c, 2b-x, b-a$ are in H.P., then $a-(x/2), b-(x/2), c-(x/2)$ are in G.P.

Option C ($\frac{20}{11}$) is correct because the series can be expressed as a telescoping series after simplifying the general term.

Option B 12 is correct because it is found by setting up an equation relating the sum of first n natural numbers to the sum of their cubes and solving for n.

Option C (HP) is correct because if a, b, c are in AP, then 1/(1-a), 1/(1-b), 1/(1-c) are in HP, which corresponds to x, y, and z.

Option B (11) is correct because it is the value of n that makes the sum of the first 2n terms of the first A.P. equal to the sum of the first n terms of the second A.P.

Option B ($\frac{7}{29}$) is correct because the sum telescopes, leaving only the first and last terms, which simplify to the given fraction.

Option A (AP) is correct because the logarithm of terms in a GP, when raised to the same power, forms an AP.

Option C (64) is correct. The expression $\frac{{1 + n + {n^2} + ... + {n^{127}}}}{{{n^m} + 1}}$ is an integer when $m = 64$ because it allows for factorization using the difference of squares.

The correct answer is 3 because the series can be simplified by rationalizing each term, leading to a telescoping series where most terms cancel out, leaving only the first and last terms.

Option B (Assertion is correct, reason is correct; reason is not a correct explanation for assertion) is correct because while both statements are true, the reason (sum of infinite GP) doesn't explain the assertion (multiplying/dividing a GP).

Option D ($x=y=z$) is correct because if x, y, z are in AP and tan^-1(x), tan^-1(y), tan^-1(z) are also in AP, then it implies that x = y = z.

Option A ($3 n(3 n+5)$) is correct because the $n^{th}$ term of the series is the product of the $n^{th}$ terms of the two sequences, which is $3n(3n+5)$.

Option C (HP) is correct because the given condition simplifies to $\frac{{2ac}}{{a + c}} = b$, which is the condition for a, b, c to be in HP.

Option A (AP) is correct because if x, y, z are in HP, then their reciprocals are in AP, and after substitution and simplification, the given terms are also in AP.