The next term of the sequence `(1+1/2)(1+1/2)(1+1/3) (1+1/2) (1+1/3) (1+1/4)` is

Options

A. 3
B. `(1+1/5)`
C. 5
D. `(1+1/2)(1+1/3)(1+1/4)(1+1/5)`

Answer

To find the next term of the sequence `(1 + 1/2)(1 + 1/2)(1 + 1/3)(1 + 1/2)(1 + 1/3)(1 + 1/4)`, we can analyze the pattern in the sequence.

### Step-by-Step Solution:

1. **Identify the Terms in the Sequence:**
- The sequence consists of terms of the form `(1 + 1/n)`, where `n` is a positive integer.
- The given terms are:
- First term: \(1 + \frac{1}{2} = \frac{3}{2}\)
- Second term: \(1 + \frac{1}{2} = \frac{3}{2}\)
- Third term: \(1 + \frac{1}{3} = \frac{4}{3}\)
- Fourth term: \(1 + \frac{1}{2} = \frac{3}{2}\)
- Fifth term: \(1 + \frac{1}{3} = \frac{4}{3}\)
- Sixth term: \(1 + \frac{1}{4} = \frac{5}{4}\)

2. **Recognize the Pattern:**
- The sequence appears to alternate and repeat certain terms:
- The term \(1 + \frac{1}{2}\) appears twice.
- The term \(1 + \frac{1}{3}\) appears twice.
- The term \(1 + \frac{1}{4}\) appears once.
- The next term should logically continue this pattern.

3. **Determine the Next Term:**
- Following the established pattern, the next term should be \(1 + \frac{1}{5}\).
- Calculate \(1 + \frac{1}{5}\):
- \(1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5}\).

4. **Conclusion:**
- The next term in the sequence is \(1 + \frac{1}{5} = \frac{6}{5}\).

### Final Answer:
The next term of the sequence is \(1 + \frac{1}{5} = \frac{6}{5}\).

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