Sum polynomial and exponential increments

A sequence of real numbers u1,u2,u3,…u_1,u_2,u_3,\ldots satisfies

un=un−1+n+2n(n≥2).u_n=u_{n-1}+n+2^n\quad(n\ge2).

Given u1=4u_1=4, show that un=(n+a)(n+b)2+2n+1u_n=\frac{(n+a)(n+b)}2+2^{n+1}, where a,ba,b are integers to be determined. Hence determine, with reasons, the behaviour of the sequence. marks

[2015/AJC/Promo/11(b)]

Solution

Explicit formula

Unroll and sum

un=un−1+n+2nu_n=u_{n-1}+n+2^n for n≥2n\ge2, with u1=4u_1=4. Show that un=(n+a)(n+b)2+2n+1u_n=\frac{(n+a)(n+b)}2+2^{n+1} where a,ba,b are integers to be determined.

Explore this idea

Hint

  1. Write unu_n as u1u_1 plus every increment from 22 to nn.

  2. Sum the arithmetic part and the geometric part separately.

Answer · Explicit formula

a=−1,b=2a=-1,b=2 (or reversed); un=(n−1)(n+2)/2+2n+1u_n=(n-1)(n+2)/2+2^{n+1}.

Explore the idea

Each term adds n + 2ⁿ

The bars are the increments from r=2r=2 upward. Switch to the running total to see un−4u_n-4.

Upper limitr = 2 to 5
Try an example
Σ r + 2^r, r = 2 to 5
74
Number of terms
4
(N−1)(N+2)/2 + 2^(N+1) − 4
74
matches the direct total
Key idea

Summing the increments from r=2r=2 to nn gives un−4u_n-4, and the geometric part 2n+1−42^{n+1}-4 soon dominates the arithmetic part.

Ideas to try
  1. Compare the closed form with the direct total as you move the upper limit. They agree for every NN.
  2. Slide to N=9N=9: the exponential part dwarfs the linear part.
Behaviour

Describe the behaviour with reasons

Hence determine, with reasons, the behaviour of the sequence.

Explore this idea

Hint

  1. Decide whether the sequence increases or decreases, using the sign of the increment.

  2. Decide whether it converges, using the formula.

Answer · Behaviour

Strictly increasing to +∞+\infty.

Explore the idea

Increasing and unbounded

Drag the slider to change n and compare the plotted values.

Indexn = 6
Try an example
term uₙ at n = 6
148
polynomial part at n = 6
20
exponential part at n = 6
128
Key idea

A strictly increasing sequence with no upper bound diverges to infinity, here because the exponential part has no ceiling.

Ideas to try
Compare the dashed parts as nn grows. Which one controls the long-run size of unu_n?
Takeaway

Unroll, then sum

Expanding a recurrence back to its first term turns it into a sum. Identify each piece as arithmetic, geometric or telescoping and apply its formula.