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The given sequence is \( a_n = \frac{n+1}{n^2+1} \), and we need to:

1. Write the first five terms of the sequence.
2. Find the limit of the sequence as \( n \to \infty \), if it exists.

### Step 1: First Five Terms of the Sequence

We will calculate the first five terms by substituting \( n = 1, 2, 3, 4, 5 \) into the formula:

- For \( n = 1 \): 
  \[
  a_1 = \frac{1+1}{1^2+1} = \frac{2}{2} = 1
  \]
  
- For \( n = 2 \): 
  \[
  a_2 = \frac{2+1}{2^2+1} = \frac{3}{4+1} = \frac{3}{5}
  \]
  
- For \( n = 3 \): 
  \[
  a_3 = \frac{3+1}{3^2+1} = \frac{4}{9+1} = \frac{4}{10} = 0.4
  \]

- For \( n = 4 \): 
  \[
  a_4 = \frac{4+1}{4^2+1} = \frac{5}{16+1} = \frac{5}{17}
  \]

- For \( n = 5 \): 
  \[
  a_5 = \frac{5+1}{5^2+1} = \frac{6}{25+1} = \frac{6}{26} = \frac{3}{13}
  \]

Thus, the first five terms are:
\[
a_1 = 1, \quad a_2 = \frac{3}{5}, \quad a_3 = 0.4, \quad a_4 = \frac{5}{17}, \quad a_5 = \frac{3}{13}
\]

### Step 2: Finding the Limit of the Sequence

Now, let's find the limit of the sequence \( a_n = \frac{n+1}{n^2+1} \) as \( n \to \infty \).

To analyze this, consider

Write the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, explain why. The sequence is given by: a_n = (n+1)/(n^2+1), with n starting at 1.

Explore the steps to calculate the first five terms of the sequence a_n = (n+1)/(n^2+1), including detailed calculations. Learn how to find the limit of the sequence as n approaches infinity, with a clear explanation of the algebraic concepts involved. Suitable for Grades 10-12.

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