Shift an affine recurrence into a GP

The nnth term of a sequence is denoted by ana_n. The sequence is defined by

an+1=pan+q(n≥1),a_{n+1}=pa_n+q\quad(n\ge1),

where p,qp,q are constants. The first three terms are a1=0,a2=2,a3=7a_1=0,a_2=2,a_3=7.

(i) Find p,qp,q. marks

(ii) Express ana_n in terms of nn, for n≥3n\ge3. marks

[2011/TJC/Prelim/II/1)]

Total marks: 6

Solution

(i)

Fit the constants

an+1=pan+qa_{n+1}=pa_n+q with a1=0a_1=0, a2=2a_2=2, a3=7a_3=7. Find pp and qq.

Explore this idea

Hint

  1. Use a1=0a_1=0 in the first transition to get qq immediately.

  2. Use the second transition to get pp.

Answer · (i)

p=5/2,q=2p=5/2,q=2.

Explore the idea

A repelling fixed point

Start at the chosen value on the horizontal axis, go up to the curve, across to the diagonal, and repeat. Each corner of the web is the next term.

Starting valueu₁ = 0
Iterations drawn3
Try an example
u(n = 4)
19.5
u(n = 1)
0
Distance to −4/3
20.83333
Key idea

The slope 52\frac52 is greater than 11, so the fixed point −43-\frac43 repels. Terms only stay there if they begin there.

Ideas to try
  1. From a1=0a_1=0 the web climbs away: 0, 2, 7, 19.50,\ 2,\ 7,\ 19.5.
  2. Start next to the fixed point −43-\frac43 and watch the web move away from it, upward or downward.
(ii)

Shift to a geometric progression

Express ana_n in terms of nn, for n≥3n\ge3.

Explore this idea

Hint

  1. Find the fixed point l=52l+2l=\frac52l+2.

  2. Show bn=an−lb_n=a_n-l is geometric, then undo the shift.

Answer · (ii)

an=43[(5/2)n−1−1]a_n=\frac43[(5/2)^{n-1}-1].

Explore the idea

The shift turns the sequence into a pure GP

The shifted sequence bn=an+43b_n=a_n+\frac43 is geometric with ratio 52\frac52; the original is the same curve lowered by 43\frac43.

Indexn = 3
Try an example
term aₙ at n = 3
7
shifted bₙ = aₙ + 4/3 at n = 3
8.3333
Key idea

Subtracting the fixed point removes +q+q, so bnb_n is geometric and its formula lowers back to ana_n.

Ideas to try
  1. At each nn the two readouts differ by exactly 43≈1.333\frac43\approx1.333.
  2. Check that consecutive bnb_n values have ratio 2.52.5.
Takeaway

Remove the constant, then use geometry

For an+1=pan+qa_{n+1}=pa_n+q with p≠1p\ne1, the fixed point l=q1−pl=\frac q{1-p} turns an−la_n-l into a geometric progression with ratio pp.