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Probability : n tosses of a fair coin no run of 3 consecutive heads appears

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Let Qn denote the probability that in n tosses of a fair coin no run of 3 consecutive heads appears. Show that:

Qn = ½ Qn-1 + ¼Qn-2 + ⅛Qn-3
Q0 = Q1 = Q2 = 1

Find Q8.

HINT: Condition of the first tail.

Please see attached for proper equation format.

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The probability that in 8 tosses of a fair coin no run of 3 consecutive heads appears is solved. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.

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Solution:
First, let us consider first three tosses in n tosses. There will be no 3 consecutive heads appear if and only if:
CASE #1: If the first coin toss results T, there must be no 3 consecutive heads appear in the next n - 1 tosses.
CASE #2: If the first coin toss results H, then the next toss must result T and there must be no 3 consecutive heads appear in the next n - 2 tosses.
CASE #3: If the first two coin tosses result HH, ...

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