find the explicit formula 1,1/2,1/3,1/4,1/5,…

To find the explicit formula for the given sequence, we first need to observe the pattern of the sequence. The given sequence is:

\[ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots \]

From this sequence, we can see that the general term for the \( n \)-th term of the sequence can be written as:

\[ a_n = \frac{1}{n} \]

Thus, the explicit formula for the \( n \)-th term of the sequence is:

\[ a_n = \frac{1}{n} \]

### Do you want details or have any questions?

Here are some related questions for further exploration:

1. What is the sum of the first \( n \) terms of this sequence?
2. How does the harmonic series, related to this sequence, behave as \( n \) approaches infinity?
3. What is the partial sum of the first \( 10 \) terms of this sequence?
4. How can this sequence be represented graphically?
5. What is the difference between the \( n \)-th term and the \((n+1)\)-th term?
6. How can you derive the \( n \)-th term of this sequence using the concept of limits?
7. What is the significance of the harmonic series in mathematical analysis?
8. Can you find another sequence with a similar pattern but different initial terms?

### Tip: 
When working with sequences, identifying the pattern or relationship between consecutive terms can often help in determining the explicit formula.

Discover the explicit formula for the sequence 1, 1/2, 1/3, 1/4, 1/5, ... with a detailed explanation and examples. Learn how to derive the formula and explore related mathematical concepts such as sequences and harmonic series. Suitable for Grades 9-12.

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