Question:
In a geometric progression of positive terms, if any term is equal to the sum of the next two terms, then the common ratio of the geometric progression is equal to
In a geometric progression of positive terms, if any term is equal to the sum of the next two terms, then the common ratio of the geometric progression is equal to
Show Hint
Always choose positive root if GP terms are positive.
Updated On: Apr 30, 2026
- $\frac{\sqrt{5}+1}{2}$
- $\frac{\sqrt{5}-1}{2}$
- $\frac{\sqrt{5}-1}{4}$
- $\frac{\sqrt{5}+1}{4}$
- $\frac{\sqrt{5}}{2}$
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
In G.P., terms are $a, ar, ar^2, \dots$
Step 1: Apply condition
\[ a = ar + ar^2 \]
Step 2: Simplify
\[ 1 = r + r^2 \Rightarrow r^2 + r - 1 = 0 \]
Step 3: Solve quadratic
\[ r = \frac{-1 \pm \sqrt{5}}{2} \] Since terms are positive, take positive root: \[ r = \frac{\sqrt{5}-1}{2} \] Final Conclusion:
Option (B)
Step 1: Apply condition
\[ a = ar + ar^2 \]
Step 2: Simplify
\[ 1 = r + r^2 \Rightarrow r^2 + r - 1 = 0 \]
Step 3: Solve quadratic
\[ r = \frac{-1 \pm \sqrt{5}}{2} \] Since terms are positive, take positive root: \[ r = \frac{\sqrt{5}-1}{2} \] Final Conclusion:
Option (B)
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM geometric progression Questions
Let a,b be two real numbers between \(3\) and \(81 \)such that the resulting sequence \(3,a,b,81\) is in a geometric progression. The value of \(a+b\) is
- If the first term of a G.P. is 3 and the sum of second and third terms is 60, then the common ratio of the G.P. is
- In a G.P., the first and third terms are 4 and 8 respectively. Then the \(21^{\text{st}}\) term is
- Let \( a_1, a_2, a_3, \ldots \) be in G.P. If \( a_1 \cdot a_2 \cdot a_3 = 64 \) and \( a_1 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5 = 32 \), then common ratio is
- The product of first 5 terms of a G.P., whose terms are increasing, is 32. The third term of the G.P. is
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions