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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 14 of 106
Marks: +1, -0
The Fibonacci sequence is defined by 1 = a1a_1 = a2a_2 and ana_n = an1+an2a_{n-1} + a_{n-2} , n > 2.
Find an+1an\frac{a_{n+1}}{a_n} , for n = 1 , 2 , 3 , 4 , 5.
Solution:  
We have, a1a_1 = 1 , a2a_2 = 1 , a3a_3 = a2+a1a_2+a_1 = 1 + 1 = 2
a4a_4 = a3+a2a_3+a_2 = 2 + 1 = 3 , a5a_5 = a4+a3a_4+a_3 = 3 + 2 = 5 , a6a_6 = a5+a4a_5+a_4 = 5 + 3 = 8
Now, substitute n = 1, 2, 3, 4, 5 in an+1an\frac{a_{n+1}}{a_n} , we obtain
a2a1\frac{a_2}{a_1} = 11\frac{1}{1} = 1 , a3a2\frac{a_3}{a_2} = 21\frac{2}{1} = 2, a4a3\frac{a_4}{a_3} = 32\frac{3}{2} , a5a4\frac{a_5}{a_4} = 53\frac{5}{3} , a6a5\frac{a_6}{a_5} = 85\frac{8}{5}.
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