JEE Mains · Maths · STD 11 - 8. sequence and series
Let \(a, b, c > 1, a^3, b^3\) and \(c^3\) be in \(A.P.\), and \(\log _a b\), \(\log _c a\) and \(\log _b c\) be in G.P. If the sum of first \(20\) terms of an \(A.P.\), whose first term is \(\frac{a+4 b+c}{3}\) and the common difference is \(\frac{a-8 b+c}{10}\) is \(-444\), then abc is equal to
- A \(343\)
- B \(216\)
- C \(\frac{343}{8}\)
- D \(\frac{125}{8}\)
Answer & Solution
Correct Answer
(B) \(216\)
Step-by-step Solution
Detailed explanation
As \(a ^3, b^3, c^3\) be in \(A.P.\) \(\rightarrow a^3+c^3=2 b^3\) \(\log _{ a }^{ b }, \log _{ c }^{ a }, \log _{ b }^{ c }\) are in \(G.P.\) \(\therefore \frac{\log b }{\log a } \cdot \frac{\log c}{\log b}=\left(\frac{\log a}{\log c}\right)^2\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right], x \in R\) and \(A^{4}=\left[a_{i j}\right] .\) If \(a_{11}=109,\) then \(a_{22}\) is equal toJEE Mains 2020 Hard
- The solution of the differential equation \(\frac{{dy}}{{dx}} = \left( {x - {y}} \right)^2\) when \(y(1) = 1\), isJEE Mains 2019 Hard
- If the value of \(\left(1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\ldots \text { upto } \infty\right)^{\log _{(0.25)}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots . \text { uptow }\right)}\) is \(l\), then \(l^{2}\) is equal to \(......\)JEE Mains 2021 Hard
- Let \(y=y(x)\) be the solution of the differential equation \(\left(3 y^2-5 x^2\right) y d x+2 x\left(x^2-y^2\right) d y=0\) such that \(y(1)=1\). then \(\left|(y(2))^3-12 y(2)\right|\) is equal to:JEE Mains 2023 Hard
- The shortest distance between the line \(y = x\) and the curve \(y^2 = x - 2\) isJEE Mains 2019 Hard
- The sum of the series \(1 + \frac{1}{{1 + 2}} + \frac{1}{{1 + 2 + 3}} + .......\) up to \(10\) terms, isJEE Mains 2013 Hard
More PYQs from JEE Mains
- \(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8 , and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5 . The probability, that \(A\) wins if A makes the first throw, isJEE Mains 2025 Medium
- The probability that two randomly selected subsets of the set \(\{1,2,3,4,5\}\) have exactly two elements in their intersection, is :JEE Mains 2021 Medium
- If the area of the region \(S=\left\{(x, y): 2 y-y^2 \leq x^2 \leq 2 y, x \geq y\right\}\) is equal to \(\frac{ n +2}{ n +1}-\frac{\pi}{ n -1}\), then the natural number \(n\) is equal to \(...............\).JEE Mains 2023 Hard
- The intercepts on \(x-\) axis made by tangents to the curve \(y = \mathop \smallint \limits_0^x \left| t \right|dt,x \in R\) which is parallel to the line \(y = 2x\) are equal to ::JEE Mains 2013 Hard
- The set of all \(\alpha\), for which the vector \(\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k} \quad\) and \(\quad \vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k} \quad\) are inclined at an obtuse angle for all \(t \in \mathbb{R}\) is :JEE Mains 2024 Hard
- Let the image of the point \(P(1, 6, a)\) in the line \(L: \dfrac{x}{1} = \dfrac{y-1}{2} = \dfrac{z-a+1}{b}\), \(b > 0\), be \(\left(\dfrac{a}{3}, 0, a+c\right)\). If \(S(\alpha, \beta, \gamma)\), \(\alpha > 0\), is the point on \(L\) such that the distance of \(S\) from the foot of perpendicular from the point \(P\) on \(L\) is \(2\sqrt{14}\), then \(\alpha + \beta + \gamma\) is equal to:JEE Mains 2026 Hard