Explanation

The sequence \(S_n = S_{n - 1} + S_{n - 2} - 1\) can be read as “to get any term in sequence S, sum the two previous terms and subtract 1.”

The problem gives the first term and the third term and asks for the fifth term:

11		10
\(S_1\)	\(S_2\)	\(S_3\)	\(S_4\)	\(S_5\)

Within the sequence \(S_1\) to \(S_3\), the problem gives two values but not the middle one (\(S_2\)). What version of the formula would include those three terms?

\(S_3 = S_2 + S_1 - 1\)

\(10 = S_2 + (11) - 1\)

\(10 = S_2 + 10\)

\(0 = S_2\)

11	0	10
\(S_1\)	\(S_2\)	\(S_3\)	\(S_4\)	\(S_5\)

To get each subsequent term, sum the two previous terms and subtract 1. Thus, \(S_4 = 10 + 0 - 1 = 9\) and \(S_5 = 9 + 10 - 1 = 18\):

11	0	10	9	18
\(S_1\)	\(S_2\)	\(S_3\)	\(S_4\)	\(S_5\)

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