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Question
Write the first 6 terms of the sequences whose nth term an is given below
an = `{{:("n", "if n is" 1"," 2 "or" 3),("a"^("n" - 1) + "a"_("n" - 2) + "a"_("n" - 3), "if n" > 3):}`
Solution
n = 1, an = n, a1 = 1
n = 2, an = n, a2 = 1
n = 3, an = n, a3 = 1
n = 4, an = an-1 + an-2 + an-3
a4 = a4-1 + a4-2 + a4-3
a4 = a3 + a2 + a1
a4 = 3 + 2 + 1 = 6
n = 5, an = an-1 + an-2 + an-3
a5 = a5-1 + a5-2 + a5-3
a5 = a4 + a3 + a2
a5 = 6 + 3 + 2 = 11
n = 6, an = an-1 + an-2 + an-3
a6 = a6-1 + a6-2 + a6-3
a6 = a5 + a4 + a3
a6 = 11 + 6 + 3 = 20
∴ The first six terms are 1, 2, 3, 6, 11, 20
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