Let \(a,b,c,d \) be an increasing sequence of real numbers,which are in geometric progression .If \(a+d=112\) and \(b+c=48\) ,then the value of \(\dfrac{a+c+8}{b}\) is
\(1\)
\(5\)
\(4\)
\(3\)
\(2\)
The Correct Option is C
Solution and Explanation
Given that:
\(a + d = 112 \)
\(b + c = 48\)
\( a + ar^3 = 112 \)
\( ⇒a (1 + r^3) = 112\)-------(1)
\( ar + ar2 = 48 \)
\(⇒a ( r + r^2 ) = 48 \)--------(2)
Now comparing \(a\) from above cases (1) and (2)we get:
\(3r^3-7r^2-7r+3=0\)
on solving we get :
\(r= -1,3,0.3334\)
Therefore from parent equation we get \(a=\dfrac{48}{12}=4\)
Then, sequence becomes \( 4, 12, 36, 108\)\(\)
Therefore , \(\dfrac{a + c +8}{ b} = \dfrac{4+36+8}{12} = 4\) (_Ans)
Top Questions on Geometric Progression
- If in a G.P. of64terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to:
- JEE Main - 2024
- Mathematics
- Geometric Progression
- In an increasing geometric progression of positive terms, the sum of the second and sixth terms is \[ \frac{70}{3} \] and the product of the third and fifth terms is 49. Then the sum of the \(4^\text{th}, 6^\text{th}\), and \(8^\text{th}\) terms is:
- JEE Main - 2024
- Mathematics
- Geometric Progression
- If the range of $f(\theta) = \frac{\sin^4\theta + 3\cos^2\theta}{\sin^4\theta + \cos^2\theta}, \, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite G.P., whose first term is $64$ and the common ratio is $\frac{\alpha}{\beta}$, is equal to ____.
- JEE Main - 2024
- Mathematics
- Geometric Progression
- Let \( a, ar, ar^2, \dots \) be an infinite G.P. If \[ \sum_{n=0}^\infty ar^n = 57 \quad \text{and} \quad \sum_{n=0}^\infty a^3 r^{3n} = 9747, \] then \( a + 18r \) is equal to:
- JEE Main - 2024
- Mathematics
- Geometric Progression
- Let \(α\) and \(β\) the roots of equation \(px^2 + qx - r = 0\), where \(P≠ 0\). If \(p,q,r\) be the consecutive term of non constant G.P and \(\frac{1}{α} + \frac{1}{β} = \frac{3}{4}\) then the value of \((α - β)^2\) is:
- JEE Main - 2024
- Mathematics
- Geometric Progression
Questions Asked in KEAM exam
- A Spherical ball is subjected to a pressure of 100 atmosphere. If the bulk modulus of the ball is 1011 N/m2,then change in the volume is:
- KEAM - 2023
- mechanical properties of solids
- Two identical solid spheres,each of radius 10cm,are kept in contact. If the mass of inertia of this system about the tangent passing through the point of contact is 0. Then mass of each sphere is:
- KEAM - 2023
- System of Particles & Rotational Motion
- The value of a(≠0) for which the equation \(\frac{1}{2}(x-2)^2+1=\sin(\frac{a}{x})\) holds is/are
- KEAM - 2023
- Trigonometric Functions
- \(∫xlog(1+x^2)dx=\)?
- KEAM - 2023
- Integration by Parts
- A uniform thin rod of mass 3kg has a length of 1m. If a point mass of 1kg is attached to it at a distance of 40cm from its center,the center of mass shifts by a distance of:
- KEAM - 2023
- System of Particles & Rotational Motion
Concepts Used:
Geometric Progression
What is Geometric Sequence?
A geometric progression is the sequence, in which each term is varied by another by a common ratio. The next term of the sequence is produced when we multiply a constant to the previous term. It is represented by: a, ar1, ar2, ar3, ar4, and so on.
Properties of Geometric Progression (GP)
Important properties of GP are as follows:
- Three non-zero terms a, b, c are in GP if b2 = ac
- In a GP,
Three consecutive terms are as a/r, a, ar
Four consecutive terms are as a/r3, a/r, ar, ar3 - In a finite GP, the product of the terms equidistant from the beginning and the end term is the same that means, t1.tn = t2.tn-1 = t3.tn-2 = …..
- If each term of a GP is multiplied or divided by a non-zero constant, then the resulting sequence is also a GP with a common ratio
- The product and quotient of two GP’s is again a GP
- If each term of a GP is raised to power by the same non-zero quantity, the resultant sequence is also a GP.
If a1, a2, a3,… is a GP of positive terms then log a1, log a2, log a3,… is an AP (arithmetic progression) and vice versa