14. The Fibonacci sequence is defined by1 - a, a, and aa 1+a 2, n> 2n-2n+1forn-1,2, 3, 4,5

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Solution

The given Fibonacci sequence is a 1 = a 2 =1 and a n = a n−1 + a n−2 . The condition is n>2.

Substitute the value of n=3 in a n = a n−1 + a n−2 .

a 3 = a 2 + a 1

Substitute the value of a 1 and a 2 in the above expression.

a 3 =1+1 =2

Similarly, substitute the value of n=4 in a n = a n−1 −1.

a 4 = a 3 + a 2

Substitute the value of a 2 and a 3 in the above expression.

a 4 =2+1 =3

Similarly, substitute the value of n=5 in a n = a n−1 −1.

a 5 = a 4 + a 3

Substitute the value of a 3 and a 4 in the above expression.

a 4 =3+2 =5

Similarly, substitute the value of n=6 in a n = a n−1 −1.

a 6 = a 5 + a 4

Substitute the value of a 3 and a 4 in the above expression.

a 6 =5+3 =8

Let r n be the ratio of the term a n+1 with respect to a n .

r n = a n+1 a n

Substitute the value of n=1 in r n .

r 1 = a 2 a 1

Substitute the values of a 2 and a 1 in the above expression.

r 1 = 1 1 =1

Substitute the value of n=2 in r n .

r 2 = a 3 a 2

Substitute the values of a 3 and a 2 in the above expression.

r 2 = 2 1 =2

Substitute the value of n=3 in r n .

r 3 = a 4 a 3

Substitute the values of a 3 and a 4 in the above expression.

r 3 = 3 2

Substitute the value of n=4 in r n .

r 4 = a 5 a 4

Substitute the values of a 4 and a 5 in the above expression.

r 4 = 5 3

Substitute the value of n=5 in r n .

r 5 = a 6 a 5

Substitute the values of a 4 and a 5 in the above expression.

r 5 = 8 5

Thus, the corresponding values of a n+1 a n are 1,2, 3 2 , 5 3 and 8 5 .

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