If an adjacaent sum sequence is geometric, must the original sequence also be geometric?

If the adjacent sum sequence of a given sequence is geometric, it does **not necessarily** imply that the original sequence is geometric.

### Here's why:

1. **Definitions**:
   - **Original sequence**: \(a_1, a_2, a_3, \dots \)
   - **Adjacent sum sequence**: Formed by adding consecutive terms of the original sequence. Let the adjacent sum sequence be \(b_1, b_2, b_3, \dots \), where:
   \[
   b_n = a_n + a_{n+1}
   \]
   The adjacent sum sequence is geometric if:
   \[
   \frac{b_{n+1}}{b_n} = r \quad \text{(constant ratio)}
   \]
   
2. **Example of non-geometric original sequence**:
   Consider the original sequence \(a_n = 2^n\), which is **exponential** (not geometric). The adjacent sum sequence becomes:
   \[
   b_n = 2^n + 2^{n+1} = 2^n(1 + 2) = 3 \cdot 2^n
   \]
   This is a geometric sequence with a constant ratio of 2. However, the original sequence \(a_n = 2^n\) is exponential, not geometric.

Thus, an adjacent sum sequence being geometric does **not imply** the original sequence must be geometric.

Would you like more details on any specific part of this explanation?

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Here are five related questions:
1. How do you prove whether a sequence is geometric?
2. Can an original sequence be arithmetic if its adjacent sum sequence is geometric?
3. How would the adjacent difference sequence behave if the original sequence is geometric?
4. What happens if the adjacent product sequence is geometric? Does the same logic hold?
5. How do sequences that are both arithmetic and geometric behave?

**Tip**: Always consider counterexamples when determining whether a condition is necessary or sufficient in mathematics!

If an adjacent sum sequence is geometric, must the original sequence also be geometric?

Explore the relationship between an original sequence and its adjacent sum sequence. This problem demonstrates why a geometric adjacent sum sequence does not necessarily mean the original sequence is geometric, supported by an example using exponential sequences. Perfect for high school students in Grades 10-12.

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