Question

If a₁, a₂, a₃ ...... are positve numbers in G.P., then which of the following is/are true -

• A. $a_1^3, a_2^3, a_3^3, ......$ are in G.P.
• B. $\frac{a_2}{a_1}, (\frac{a_3}{a_2})^2, (\frac{a_4}{a_3})^3 ......$ are in G.P.
• C. $lna_1, lna_2, lna_3 ......$ are in A.P.
• D. $a_1a_2, a_2a_3, a_3a_4 ......$ are in G.P.

Answer 1

Answer: (A), (B), (C), (D)

All four statements are true.

Answer 2

Let the given geometric progression (G.P.) be $a_1, a_2, a_3, \dots$. Let $a_1$ be the first term and $r$ be the common ratio. Since $a_i$ are positive numbers, $a_1 > 0$ and $r > 0$. The general term of the G.P. is $a_n = a_1 r^{n-1}$.

1. $a_1^3, a_2^3, a_3^3, ......$ are in G.P.

The terms of this sequence are $b_n = a_n^3$. Substitute $a_n = a_1 r^{n-1}$: $b_n = (a_1 r^{n-1})^3 = a_1^3 (r^{n-1})^3 = a_1^3 r^{3(n-1)}$. This is in the form $A \cdot R^{n-1}$, where $A = a_1^3$ and $R = r^3$. Thus, $a_1^3, a_2^3, a_3^3, \dots$ is a G.P. with first term $a_1^3$ and common ratio $r^3$. This statement is TRUE.

2. $\frac{a_2}{a_1}, (\frac{a_3}{a_2})^2, (\frac{a_4}{a_3})^3 ......$ are in G.P.

For a G.P., the ratio of consecutive terms is the common ratio $r$. So, $\frac{a_{n+1}}{a_n} = r$. Let the terms of this new sequence be $c_n$.

$c_1 = \frac{a_2}{a_1} = r$

$c_2 = (\frac{a_3}{a_2})^2 = r^2$

$c_3 = (\frac{a_4}{a_3})^3 = r^3$

In general, $c_n = (\frac{a_{n+1}}{a_n})^n = r^n$. The sequence is $r, r^2, r^3, \dots$. To check if it's a G.P., we find the ratio of consecutive terms: $\frac{c_{n+1}}{c_n} = \frac{r^{n+1}}{r^n} = r$. Since the ratio is constant ($r$), this sequence is a G.P. with first term $r$ and common ratio $r$. This statement is TRUE.

3. $lna_1, lna_2, lna_3 ......$ are in A.P.

The terms of this sequence are $d_n = lna_n$. Substitute $a_n = a_1 r^{n-1}$: $d_n = ln(a_1 r^{n-1})$. Using logarithm properties, $ln(xy) = lnx + lny$ and $ln(x^y) = ylnx$: $d_n = lna_1 + ln(r^{n-1}) = lna_1 + (n-1)lnr$. This is in the form $A + (n-1)D$, where $A = lna_1$ and $D = lnr$. Thus, $lna_1, lna_2, lna_3, \dots$ is an A.P. with first term $lna_1$ and common difference $lnr$. Since $a_i > 0$, $lna_i$ are well-defined. This statement is TRUE.

4. $a_1a_2, a_2a_3, a_3a_4 ......$ are in G.P.

The terms of this sequence are $e_n = a_n a_{n+1}$. Substitute $a_n = a_1 r^{n-1}$ and $a_{n+1} = a_1 r^n$: $e_n = (a_1 r^{n-1})(a_1 r^n) = a_1^2 r^{n-1+n} = a_1^2 r^{2n-1}$. We can rewrite this as $a_1^2 r^{2(n-1)+1} = a_1^2 r \cdot (r^2)^{n-1}$. This is in the form $A \cdot R^{n-1}$, where $A = a_1^2 r$ and $R = r^2$. Thus, $a_1a_2, a_2a_3, a_3a_4, \dots$ is a G.P. with first term $a_1^2r$ and common ratio $r^2$. This statement is TRUE.

All four statements are true.