If p, q, r are in one geometric progression and a, b, c are in another geometric progression, then ap, bq, cr are in
Geometric progression
Understanding Geometric Progressions
This problem deals with the properties of geometric progressions (GPs). A sequence of numbers is called a geometric progression if the ratio of any term to its preceding term is constant. This constant ratio is known as the common ratio.
Analyzing the Given Geometric Progressions
We are given two separate geometric progressions:
- The first GP has terms $p, q, r$. Let the common ratio of this GP be $r_1$.
- The second GP has terms $a, b, c$. Let the common ratio of this GP be $r_2$.
Based on the definition of a geometric progression, we can write the terms in relation to their common ratios:
- For the first GP ($p, q, r$):
- $q = p \cdot r_1$
- $r = q \cdot r_1 = (p \cdot r_1) \cdot r_1 = p \cdot r_1^2$
- For the second GP ($a, b, c$):
- $b = a \cdot r_2$
- $c = b \cdot r_2 = (a \cdot r_2) \cdot r_2 = a \cdot r_2^2$
So, the terms of the first GP are $p, p \cdot r_1, p \cdot r_1^2$, and the terms of the second GP are $a, a \cdot r_2, a \cdot r_2^2$.
Forming the New Terms: ap, bq, cr
We are asked to determine the type of progression formed by the terms $ap, bq, cr$. Let's express these new terms using the relationships we found:
- The first new term is $ap$.
- The second new term is $bq = (a \cdot r_2) \cdot (p \cdot r_1) = ap \cdot r_1 \cdot r_2$.
- The third new term is $cr = (a \cdot r_2^2) \cdot (p \cdot r_1^2) = ap \cdot r_1^2 \cdot r_2^2$.
So, the sequence of new terms is $ap, ap \cdot r_1 r_2, ap \cdot (r_1 r_2)^2$.
Checking the Progression Type for ap, bq, cr
To check if these new terms form a Geometric Progression, we need to see if the ratio of consecutive terms is constant.
Ratio of the second term to the first term:
$\frac{bq}{ap} = \frac{ap \cdot r_1 r_2}{ap} = r_1 r_2$
(Assuming $ap \neq 0$. If $a$ or $p$ is zero, the sequence might be trivial, but generally, GP terms are non-zero).
Ratio of the third term to the second term:
$\frac{cr}{bq} = \frac{ap \cdot r_1^2 r_2^2}{ap \cdot r_1 r_2} = \frac{r_1^2 r_2^2}{r_1 r_2} = r_1 r_2$
Since the ratio of the second term to the first term ($r_1 r_2$) is equal to the ratio of the third term to the second term ($r_1 r_2$), the new terms $ap, bq, cr$ form a geometric progression.
The common ratio of this new geometric progression is $r_1 r_2$, which is the product of the common ratios of the two original GPs.
Summary of Terms and Ratios
| Sequence | Terms | Common Ratio |
|---|---|---|
| First GP | $p, q=pr_1, r=pr_1^2$ | $r_1$ |
| Second GP | $a, b=ar_2, c=ar_2^2$ | $r_2$ |
| New Sequence | $ap, bq=ap(r_1r_2), cr=ap(r_1r_2)^2$ | $r_1r_2$ |
The analysis clearly shows that $ap, bq, cr$ follow the definition of a geometric progression with a common ratio $r_1 r_2$.
Revision Table: Types of Progressions
| Progression Type | Definition | Condition |
|---|---|---|
| Arithmetic Progression (AP) | Common difference between consecutive terms. | $b-a = c-b$ (or $a+c = 2b$) for terms $a, b, c$. |
| Geometric Progression (GP) | Common ratio between consecutive terms. | $\frac{b}{a} = \frac{c}{b}$ (or $b^2 = ac$) for terms $a, b, c$. |
| Harmonic Progression (HP) | Reciprocals of terms are in AP. | $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in AP, meaning $\frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}$. |
Additional Information on Geometric Progressions
A geometric progression is a fundamental sequence in mathematics. Here are some key points:
- General Form: A GP can be written as $A, AR, AR^2, AR^3, \dots$, where $A$ is the first term and $R$ is the common ratio.
- $n^{th}$ Term: The $n^{th}$ term of a GP is given by $a_n = A \cdot R^{n-1}$.
- Sum of $n$ Terms: The sum of the first $n$ terms of a GP is $S_n = A \frac{1-R^n}{1-R}$ (when $R \neq 1$). If $R=1$, $S_n = nA$.
- Infinite Sum: If $|R| < 1$, the sum of an infinite GP converges to $S_\infty = \frac{A}{1-R}$.
- Geometric Mean: For two numbers $x$ and $y$, their geometric mean is $\sqrt{xy}$. For three numbers $a, b, c$ to be in GP, $b$ is the geometric mean of $a$ and $c$ (i.e., $b = \sqrt{ac}$, or $b^2 = ac$). This was the condition we verified for $ap, bq, cr$.
Understanding how operations like multiplication of corresponding terms affect progressions is important in sequences and series problems.
Important Questions from Geometric Progressions
If \(2^{\frac{1}{c}}, 2^{\frac{b}{a c}}, 2^{\frac{1}{a}}\) are in GP, then which one of the following is correct ?
If G is the geometric mean of numbers 1, 2, 22, 23,.....2n-1, then what is the value of 1 + 2log2G ?
If m is the geometric mean of \({\left( {\frac{{\rm{y}}}{{\rm{z}}}} \right)^{\log \left( {{\rm{yz}}} \right)}},{\rm{\;}}{\left( {\frac{{\rm{z}}}{{\rm{x}}}} \right)^{\log \left( {{\rm{zx}}} \right)}}{\rm{\;and\;}}{\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)^{\log \left( {{\rm{xy}}} \right)}}\) then what is the value of m?
The value of the infinite product \({6^{\frac{1}{2}}} \times {6^{\frac{1}{2}}} \times {6^{\frac{3}{8}}} \times {6^{\frac{1}{4}}} \times \ldots \) is
The geometric mean of the observations x 1, x 2, x 3, … x nis G 1. The geometric mean of the observations y 1, y 2, y 3,… y nis G 2. The geometric mean of observations \(\frac{{{{\rm{x}}_1}}}{{{{\rm{y}}_1}}},\frac{{{{\rm{x}}_2}}}{{{{\rm{y}}_2}}},\frac{{{{\rm{x}}_3}}}{{{{\rm{y}}_3}}}, \ldots \frac{{{{\rm{x}}_{\rm{n}}}}}{{{{\rm{y}}_{\rm{n}}}}}\) is
