Nomsa generates a sequence which is both arithmetic and geometric. The first term
is 1 . She claims that there is only one such sequence. Is that correct? Show ALL
your workings to justify your answer.

Question

UpStudy AI · Accepted Answer

Yes, Nomsa is correct. There is only one sequence that is both arithmetic and geometric with the first term being 1. Let's explore why this is the case by examining the properties of arithmetic and geometric sequences.

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### **Definitions**

1. **Arithmetic Sequence**: A sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the **common difference** $$ d $$.

$$
   a_{n} = a_{1} + (n - 1)d
   $$
   
2. **Geometric Sequence**: A sequence of numbers where the ratio between consecutive terms is constant. This constant ratio is called the **common ratio** $$ r $$.

$$
   a_{n} = a_{1} \times r^{(n-1)}
   $$

---

### **Given**

- The first term $$ a_{1} = 1 $$.
- The sequence is **both** arithmetic and geometric.

---

### **Objective**

Determine if there exists more than one sequence that satisfies the above conditions. Show all steps to justify the conclusion.

---

### **Solution**

1. **Expressing the Second Term**

Let's denote the second term of the sequence as $$ a_{2} $$.

- **From the Arithmetic Sequence**:
     
     $$
     a_{2} = a_{1} + d = 1 + d
     $$
   
   - **From the Geometric Sequence**:
     
     $$
     a_{2} = a_{1} \times r = 1 \times r = r
     $$
     
   Since $$ a_{2} $$ must satisfy both the arithmetic and geometric properties, we set the two expressions equal to each other:

$$
   1 + d = r \quad \text{(Equation 1)}
   $$

2. **Expressing the Third Term**

Let's now consider the third term $$ a_{3} $$.

- **From the Arithmetic Sequence**:
     
     $$
     a_{3} = a_{2} + d = (1 + d) + d = 1 + 2d
     $$
   
   - **From the Geometric Sequence**:
     
     $$
     a_{3} = a_{2} \times r = r \times r = r^{2}
     $$
     
   Again, setting the two expressions equal:

$$
   1 + 2d = r^{2} \quad \text{(Equation 2)}
   $$

3. **Substituting Equation 1 into Equation 2**

From Equation 1, we have $$ r = 1 + d $$. Substitute this into Equation 2:

$$
   1 + 2d = (1 + d)^{2}
   $$
   
   Expanding the right side:

$$
   1 + 2d = 1 + 2d + d^{2}
   $$
   
4. **Simplifying the Equation**

Subtract $$ 1 + 2d $$ from both sides to isolate $$ d^{2} $$:

$$
   (1 + 2d) - (1 + 2d) = d^{2} \
   0 = d^{2}
   $$
   
   Solving for $$ d $$:

$$
   d = 0
   $$
   
5. **Determining the Common Ratio $$ r $$**

Substitute $$ d = 0 $$ back into Equation 1:

$$
   r = 1 + d = 1 + 0 = 1
   $$
   
6. **Constructing the Sequence**

With $$ d = 0 $$ and $$ r = 1 $$, the sequence can be described as follows:

- **Arithmetic View**:
     
     $$
     a_{n} = 1 + (n - 1) \times 0 = 1
     $$
   
   - **Geometric View**:
     
     $$
     a_{n} = 1 \times 1^{(n-1)} = 1
     $$
   
   Therefore, every term in the sequence is:

$$
   1, \, 1, \, 1, \, 1, \, \dots
   $$
   
   This is a **constant sequence** where each term is identical.

---

### **Conclusion**

The only sequence that is both **arithmetic** and **geometric** with the first term $$ a_{1} = 1 $$ is the **constant sequence** where every term is 1. There are no other sequences that satisfy both properties simultaneously under these conditions. Thus, Nomsa's claim is correct.
There is only one such sequence: a constant sequence where every term is 1.

Nomsa generates a sequence which is both arithmetic and geometric. The first term
is 1 . She claims that there is only one such sequence. Is that correct? Show ALL
your workings to justify your answer.

Upstudy AI Solution

Answer

Solution

Definitions

Given

Objective

Solution

Conclusion

Beyond the Answer

Related Questions

Latest Algebra Questions

Nomsa generates a sequence which is both arithmetic and geometric. The first term is 1 . She claims that there is only one such sequence. Is that correct? Show ALL your workings to justify your answer.

Upstudy AI Solution

Answer

Solution

Definitions

Given

Objective

Solution

Conclusion

Beyond the Answer

Related Questions

Latest Algebra Questions

Nomsa generates a sequence which is both arithmetic and geometric. The first term
is 1 . She claims that there is only one such sequence. Is that correct? Show ALL
your workings to justify your answer.