Suppose \(W_1 = 1, W_2 = 2\) and for \(n>2,\, W_n = W_1 + W_2 + ... + W_{n−2} + 2\). Find the values of \(W_1, W_2, ..., W_8\). Do you recognise this sequence?

SOLUTION

Write down what is given

\(W_1 = 1, W_2 = 2\)
\(W_n = W_1 + W_2 + ... + W_{n−2} + 2\)

Find the values

\(W_3 = 1 + 2 = 3\)
\(W_4 = 1 + 2 + 2 = 5\)
\(W_5 = 1 + 2 + 3 + 2 = 8\)
\(W_6 = 1 + 2 + 3 + 5 + 2 = 13\)
\(W_7 = 1 + 2 + 3 + 5 + 8 + 2 = 21\)
\(W_8 = 1 + 2 + 3 + 5 + 8 + 13 + 2 = 34\)

Recognise the pattern

1, 2, 3, 5, 8, 13, 21, 34,... is the famous Fibonacci Sequence, where each term equals the sum of the previous two terms.

Concept Note

Many recursive sequences can be rewritten into a simpler recurrence relation. The Fibonacci sequence is characterised by:
\(F_n = F_{n−1} + F_{n−2}\),
with starting terms \(1, 2\).