12.a,--1, an, n2

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Solution

The Fibonacci sequence is given by

1= a 1 = a 2

The n th term of the sequence is given as,

a n = a n−1 + a n−2 (1)

Substitute n=3 in equation (1),

a 3 = a 3−1 + a 3−2 = a 2 + a 1 =1+1 =2

Substitute n=4 in equation (1),

a 4 = a 4−1 + a 4−2 = a 3 + a 2 =2+1 =3

Substitute n=5 in equation (1),

a 5 = a 5−1 + a 5−2 = a 4 + a 3 =3+2 =5

Substitute n=6 in equation (1),

a 6 = a 6−1 + a 6−2 = a 5 + a 4 =5+3 =8

For n=1,

a n+1 a n = a 1+1 a 1 = a 2 a 1 = 1 1 =1

For n=2,

a n+1 a n = a 2+1 a 2 = a 3 a 2 = 2 1 =2

For n=3,

a n+1 a n = a 3+1 a 3 = a 4 a 3 = 3 2

For n=4,

a n+1 a n = a 4+1 a 4 = a 5 a 4 = 5 3

For n=5,

a n+1 a n = a 5+1 a 5 = a 6 a 5 = 8 5

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