Q.The sum or difference of two G.P.s, is again a G.P.

A Geometric Progression (G.P.) is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Consider two G.P.s:

1. $G_1 = \{1, 2, 4, 8, \dots\}$ with first term $a_1 = 1$ and common ratio $r_1 = 2$.
2. $G_2 = \{1, 3, 9, 27, \dots\}$ with first term $a_2 = 1$ and common ratio $r_2 = 3$.

Let's find the sum of these two G.P.s term by term:

$S = G_1 + G_2 = \{1+1, 2+3, 4+9, 8+27, \dots\} = \{2, 5, 13, 35, \dots\}$.

To check if $S$ is a G.P., we examine the ratios of consecutive terms:

The ratio of the second term to the first term is $\frac{5}{2}$. …

A Geometric Progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For a sequence to be a G.P., this common ratio must be constant throughout the sequence.

The statement claims that if you take two G.P.s and either add or subtract their corresponding terms, the resulting sequence will always be another G.P. Let's explore why this is generally not true by considering the fundamental property of a G.P.

Why the Statement is Generally False

For a sequence to be a G.P., the ratio of any term to its preceding term must be constant. When we add or subtract terms from two different G.P.s, especially those with different common ratios, the resulting sequence typically loses this constant ratio property.

Let's demonstrate this with a concrete example.

1. Define two distinct G.P.s.

   Consider two G.P.s, $A$ and $B$, with their respective first terms and common ratios:

   • G.P. $A$: First term $a_1 = 1$, common ratio $r_A = 2$. The terms are $1, 2, 4, 8, 16, \dots$ The $n$-th term is $A_n = 1 \cdot 2^{n-1} = 2^{n-1}$.
   • G.P. $B$: First term $b_1 = 1$, common ratio $r_B = 3$. The terms are $1, 3, 9, 27, 81, \dots$ The $n$-th term is $B_n = 1 \cdot 3^{n-1} = 3^{n-1}$.

2. Form a new sequence by summing the terms.

   Let's create a new sequence, $C$, by adding the corresponding terms of G.P. $A$ and G.P. $B$. The $n$-th term of sequence $C$ will be $C_n = A_n + B_n$.

   • $C_1 = A_1 + B_1 = 1 + 1 = 2$
   • $C_2 = A_2 + B_2 = 2 + 3 = 5$
   • $C_3 = A_3 + B_3 = 4 + 9 = 13$
   • $C_4 = A_4 + B_4 = 8 + 27 = 35$
   • $C_5 = A_5 + B_5 = 16 + 81 = 97$ The sequence $C$ is $2, 5, 13, 35, 97, \dots$

3. Check if the new sequence $C$ is a G.P.

   For $C$ to be a G.P., the ratio of consecutive terms must be constant. Let's calculate these ratios:

   • Ratio of $C_2$ to $C_1$: $\frac{C_2}{C_1} = \frac{5}{2} = 2.5$
   • Ratio of $C_3$ to $C_2$: $\frac{C_3}{C_2} = \frac{13}{5} = 2.6$
   • Ratio of $C_4$ to $C_3$: $\frac{C_4}{C_3} = \frac{35}{13} \approx 2.69$
   • Ratio of $C_5$ to $C_4$: $\frac{C_5}{C_4} = \frac{97}{35} \approx 2.77$

   Since $\frac{5}{2} \neq \frac{13}{5} \neq \frac{35}{13} \neq \frac{97}{35}$, the ratios are not constant. Therefore, the sequence $C$ is not a G.P.

   The same logic applies to the difference of two G.P.s. If we consider $D_n = A_n - B_n$:

   • $D_1 = 1 - 1 = 0$
   • $D_2 = 2 - 3 = -1$
   • $D_3 = 4 - 9 = -5$
   • $D_4 = 8 - 27 = -19$ The sequence $D$ is $0, -1, -5, -19, \dots$. …