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TMUA 2017 · Paper 2 · Question 6 of 20

TMUA 2017 Paper 2 Question 6

Differentiation and integration — A sequence defined by a definite integral. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 2Differentiation and integrationA sequence defined by a definite integral8 options
A sequence u0, u1, u2, … is defined as follows: u0 = 1 un = ∫01 4x un−1 dx   for  n ≥ 1 What is the value of u1000?
  1. A21000
  2. B41000
  3. C41000!
  4. D41001!
  5. E210001000!
  6. F410001000!
  7. G210001001!
  8. H410001001!
Show the answer and worked solution
answer · A
  1. A21000
  2. B41000
  3. C41000!
  4. D41001!
  5. E210001000!
  6. F410001000!
  7. G210001001!
  8. H410001001!
The point to notice is that un−1 is a number, not a function of x, so it comes straight out of the integral. That leaves un = un−1∫01 4x dx = un−1[2x2]01 = 2un−1. Starting from u0 = 1, each step doubles, so un = 2n and u1000 = 21000. The factorials in the other options are what you would get if you wrongly treated the integral as building up a new power of x each time.