Let \(a,b,c>1,a^3,b^3 \text{ and } c^3\) be in A.P., and \(log_ab, log_ca \text{ and } log_bc\) be in G.P. If the sum of first 20 terms of an A.P., whose first term is \(\frac{a+4b+c}{3}\) and the common difference is \(\frac{a−8b+c}{10}\) is −444, then abc is equal to:
- 343
- 216
- $\frac{343}{8}$
- $\frac{125}{8}$
The Correct Option is B
Solution and Explanation
are in G.P.
......(2)
From (1) and (2)
Top Questions on nth Term of an AP
- What is the sum of the infinite series \( S = \sum_{n=0}^{\infty} \frac{1}{3^n} \)?
- JEE Main - 2025
- Mathematics
- nth Term of an AP
Let $a_1, a_2, a_3, \ldots$ be an AP If $a_7=3$, the product $a_1 a_4$ is minimum and the sum of its first $n$ terms is zero, then $n !-4 a_{n(n+2)}$ is equal to :
- JEE Main - 2023
- Mathematics
- nth Term of an AP
Let $a_1, a_2, \ldots, a_n$ be in AP If $a_5=2 a_7$ and $a_{11}=18$, then $12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots+\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$ is equal to
- JEE Main - 2023
- Mathematics
- nth Term of an AP
- The $8^{\text {th }}$ common term of the series
$S_1=3+7+11+15+19+\ldots \ldots$
$S_2=1+6+11+16+21+\ldots $ is ______- JEE Main - 2023
- Mathematics
- nth Term of an AP
- Let $a, b, c>, a^3, b^3$ and $c^3$ be in AP, and $\log _a b, \log _c a$ and $\log _b c$ be in GP If the sum of first 20 terms of an AP, whose first term is $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$ is $-444$, then $a b c$ is equal to:
- JEE Main - 2023
- Mathematics
- nth Term of an AP
Questions Asked in JEE Main exam
Statement-1: \( \text{ClF}_3 \) has 3 possible structures.
Statement-2: \( \text{III} \) is the most stable structure due to least lone pair-bond pair (lp-bp) repulsion.
Which of the following options is correct?
- JEE Main - 2025
- Molecular Stability
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged.
Reason (R): By using the choke coil, the voltage across the tube is reduced by a factor \( \frac{R}{\sqrt{R^2 + \omega^2 L^2}} \), where \( \omega \) is the frequency of the supply across resistor \( R \) and inductor \( L \). If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage.
In light of the above statements, choose the most appropriate answer from the options given below:
- JEE Main - 2025
- Electromagnetism
- Let \( (2, 3) \) be the largest open interval in which the function \( f(x) = 2 \log_e (x - 2) - x^2 + ax + 1 \) is strictly increasing, and \( (b, c) \) be the largest open interval, in which the function \( g(x) = (x - 1)^3 (x + 2 - a)^2 \) is strictly decreasing. Then \( 100(a + b - c) \) is equal to:
- JEE Main - 2025
- Quadratic Equations
- If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
- JEE Main - 2025
- Complex numbers
- If \( \alpha \) and \( \beta \) are the roots of the equation \( 2z^2 - 3z - 2i = 0 \), where \( i = \sqrt{-1} \), then \[ 16 \cdot {Re} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \cdot {Im} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \] is equal to:
- JEE Main - 2025
- Complex numbers
Concepts Used:
Sequence and Series
Sequence: Sequence and Series is one of the most important concepts in Arithmetic. A sequence refers to the collection of elements that can be repeated in any sort.
Eg: a1,a2,a3, a4…….
Series: A series can be referred to as the sum of all the elements available in the sequence. One of the most common examples of a sequence and series would be Arithmetic Progression.
Eg: If a1,a2,a3, a4……. etc is considered to be a sequence, then the sum of terms in the sequence a1+a2+a3+ a4……. are considered to be a series.
Types of Sequence and Series:
Arithmetic Sequences
A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence.
Geometric Sequences
A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.
Harmonic Sequences
A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.
Fibonacci Numbers
Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2