Source: http://nrich.maths.org/475 Loopy A Fractional Sequence Source: http://nrich.maths.org/475 Loopy

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =3, 𝑎 2 =7 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture?

𝑎 3 = 1+𝑏 𝑎 A Fractional Sequence 𝒌 𝒂 𝒌 2 𝑏 3 4 5 6 7 8 9 10 11 12 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏 𝑎 3 = 1+𝑏 𝑎

𝑎 4 = 1+ 1+𝑏 𝑎 𝑏 = 1+𝑎+𝑏 𝑎𝑏 A Fractional Sequence 𝒌 𝒂 𝒌 2 𝑏 3 1+𝑏 𝑎 4 5 6 7 8 9 10 11 12 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏 𝑎 4 = 1+ 1+𝑏 𝑎 𝑏 = 1+𝑎+𝑏 𝑎𝑏

𝑎 5 = 1+ 1+𝑎+𝑏 𝑎𝑏 1+𝑏 𝑎 = 𝑎𝑏+1+𝑎+𝑏 𝑎𝑏 1+𝑏 𝑎 = 1+𝑎 1+𝑏 𝑏 1+𝑏 = 1+𝑎 𝑏 A Fractional Sequence 𝒌 𝒂 𝒌 1 𝑎 2 𝑏 3 1+𝑏 𝑎 4 1+𝑎+𝑏 𝑎𝑏 5 6 7 8 9 10 11 12 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏 𝑎 5 = 1+ 1+𝑎+𝑏 𝑎𝑏 1+𝑏 𝑎 = 𝑎𝑏+1+𝑎+𝑏 𝑎𝑏 1+𝑏 𝑎 = 1+𝑎 1+𝑏 𝑏 1+𝑏 = 1+𝑎 𝑏

𝑎 6 = 1+ 1+𝑎 𝑏 1+𝑎+𝑏 𝑎𝑏 = 1+𝑎+𝑏 𝑏 1+𝑎+𝑏 𝑎𝑏 =𝑎 A Fractional Sequence 𝒌 𝒂 𝒌 1 𝑎 2 𝑏 3 1+𝑏 𝑎 4 1+𝑎+𝑏 𝑎𝑏 5 1+𝑎 𝑏 6 7 8 9 10 11 12 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏 𝑎 6 = 1+ 1+𝑎 𝑏 1+𝑎+𝑏 𝑎𝑏 = 1+𝑎+𝑏 𝑏 1+𝑎+𝑏 𝑎𝑏 =𝑎

𝑎 7 = 1+𝑎 1+𝑎 𝑏 =𝑏 A Fractional Sequence 𝒌 𝒂 𝒌 2 𝑏 3 1+𝑏 𝑎 4 1+𝑎+𝑏 𝑎𝑏 5 1+𝑎 𝑏 6 7 8 9 10 11 12 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏 𝑎 7 = 1+𝑎 1+𝑎 𝑏 =𝑏

A Fractional Sequence 𝒌 𝒂 𝒌 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏 3 1+𝑏 𝑎 4 1+𝑎+𝑏 𝑎𝑏 5 1+𝑎 𝑏 6 7 8 9 10 11 12 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏 The sequence loops and so we can go straight to the 11th and 12th terms.

A Fractional Sequence 𝒌 𝒂 𝒌 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏 3 1+𝑏 𝑎 4 1+𝑎+𝑏 𝑎𝑏 5 1+𝑎 𝑏 6 7 8 9 10 11 𝒂 12 𝒃 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏 The sequence loops and so we can go straight to the 11th and 12th terms.

RESOURCES

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =3, 𝑎 2 =7 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =4, 𝑎 2 =3 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =2, 𝑎 2 =5 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =11, 𝑎 2 =17 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =13, 𝑎 2 =8 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =17, 𝑎 2 =5 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =17, 𝑎 2 =3 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =333, 𝑎 2 =297 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =997, 𝑎 2 =21 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =9, 𝑎 2 =3 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =−2, 𝑎 2 =5 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =3, 𝑎 2 =−3 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =100, 𝑎 2 =7 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =1000, 𝑎 2 =7 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =1000, 𝑎 2 =351 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23

A Fractional Sequence The terms of a sequence are given by the inductive formula:   𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =537, 𝑎 2 =11 𝒌 𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12 Find the values of 𝑎 11 and 𝑎 12 . Experiment with your own values of 𝑎 1 and 𝑎 2 Make a conjecture about the behaviour of this sequence. Can you prove your conjecture? SIC_23