rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ and $l$ different from the colors it has in days $i$ and $k$. what is the maximum number of days rainbow can live in terms of $n$?
To determine the maximum number of days the bird can live, we need to ensure that the bird's colors on consecutive days are not the same and that there are no four days \(i, j, k, l\) such that \(i < j < k < l\) and the bird has the same color on days \(i\) and \(k\) and the same color on days \(j\) and \(l\) different from the colors on days \(i\) and \(k\).

1. **Base Case:**
   For \(n = 1\), the bird can only have one color. Therefore, the bird can live for only 1 day. This satisfies the condition since there are no consecutive days to compare.

2. **Inductive Hypothesis:**
   Assume that for \(i < n\), the bird can live for at most \(2i - 1\) days with \(i\) colors.

3. **Inductive Step:**
   We need to show that for \(n\) colors, the bird can live for at most \(2n - 1\) days.

   Suppose the bird can live for at least \(2n\) days. Let \(k\) be the color used on the first day. Let the days on which color \(k\) is used be \(D_1, D_2, \ldots, D_m\), where \(D_m\) is the last day color \(k\) is used.

4. **Intervals Analysis:**
   Consider the intervals \((D_1, D_2), (D_2, D_3), \ldots, (D_{m-1}, D_m), (D_m, D)\). Another color cannot appear in two of these intervals; otherwise, the condition would be violated.

5. **Color Usage Calculation:**
   Let \(a_i\) be the number of days in the interval \((D_i, D_{i+1})\). For a proper coloring, we need at least \(\left\lfloor \frac{a_i}{2} \right\rfloor + 1 \geq \frac{a_i + 1}{2}\) colors in each interval.

6. **Total Colors Calculation:**
   The total number of colors used is at least:
   \[
   \sum_{i=1}^{m} \left( \frac{a_i + 1}{2} \right) + 1
   \]
   Since \(\sum_{i=1}^{m} a_i = 2n - m\), we need at least \(n + 1\) colors, which is a contradiction because we only have \(n\) colors.

Therefore, the maximum number of days the bird can live is \(2n - 1\).

The final answer is \(\boxed{2n - 1}\).