Trending Articles
Shocking Minecraft Pocket Edition Mod Revealed – Transform Your Experience NOW!
A virologist is studying the combinations of synthetic DNA sequences where two types of nucleotides, A and T, are used. How many 6-nucleotide sequences can be formed where there are exactly two consecutive A's at least once?
To solve this, we first compute the total number of 6-nucleotide sequences using A and T, which is \(2^6 = 64\).
Next, we compute the number of sequences that do **not** contain two consecutive A’s. Let’s denote this count as \(a_n\) for sequences of length \(n\).
We derive a recurrence for \(a_n\):
If the first nucleotide is T, the rest \(n-1\) positions form a valid sequence: \(a_{n-1}\)
If the first is A, the next must be T (to avoid AA), and then \(n-2\) positions follow: \(a_{n-2}\)
So, \(a_n = a_{n-1} + a_{n-2}\), with base cases:
\(a_1 = 2\) (A, T)
\(a_2 = 3\) (TT, TA, AT — not AA)
