For cyclic peptides:
• Count unique sequences considering the cyclic structure to avoid duplicates.
• Use the number of amino acids and their combinations to determine possible arrangements.
4
2
1. Cyclic Tripeptides:
- A cyclic tripeptide is a peptide consisting of three amino acids arranged in a cyclic structure without any free ends.
- Given two amino acids, A and B, the tripeptide can have different arrangements depending on the sequence of A and B.
2. Possible Arrangements:
- The cyclic nature of the tripeptide eliminates linear sequence redundancy.
- The distinct arrangements are:
1. A-A-A
2. A-A-B
3. A-B-A
4. A-B-B
Total: 4 unique cyclic tripeptides.
Let \( S = \left\{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \right\} \), where
\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \]Then \( n(S) \) is equal to ______.
Let \( T_r \) be the \( r^{\text{th}} \) term of an A.P. If for some \( m \), \( T_m = \dfrac{1}{25} \), \( T_{25} = \dfrac{1}{20} \), and \( \displaystyle\sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle\sum_{r=m}^{2m} T_r \) is equal to: