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प्रश्न
The Fibonacci sequence is defined by 1 = a1 = a2 and an = an – 1 + an – 2, n > 2.
Find `a_(n+1)/a_n`, for n = 1, 2, 3, 4, 5
उत्तर
a1 = 1, a2 = 1
an = an−1 + an−2
Putting n = 3, 4, 5, 6,
a3 = a2 + a1 = 1 + 1 = 2
a4 = a3 + a2 = 2 + 1 = 3
a5 = a4 + a3 = 3 + 2 = 5
a6 = a5 + a4 = 5 + 3 = 8
Now, substituting n = 1, 2, 3, 4, 5 in `"a"_("n" + 1)/"a"_"n"`
∴ n = `1, (a_n + 1)/(a_n) = a_2/a_1 = 1/1 = 1`,
n = `2, (a_n + 1)/(a_n) = "a"_3/"a"_2 = 2/1 = 2`,
n = `3, (a_n + 1)/(a_n) = "a"_4/"a"_3 = 3/2`,
n = `4, (a_n + 1)/(a_n) = "a"_5/"a"_4 = 5/3`,
n = `5, (a_n + 1)/(a_n) = "a"_6/"a"_5 = 8/5`
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