Four terms identify a cubic

The first four terms of a sequence are given by T1=1,T2=2,T3=4,T4=8T_1=1,T_2=2,T_3=4,T_4=8. Given that TrT_r is a cubic polynomial in rr, find TrT_r in terms of rr. marks

[2011/PJC/Prelim II/1]

Solution

Q

Polynomial fitting

The first four terms of a sequence are T1=1T_1=1, T2=2T_2=2, T3=4T_3=4, T4=8T_4=8. Given that TrT_r is a cubic polynomial in rr, find TrT_r in terms of rr.

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Hint

  1. Write Tr=ar3+br2+cr+dT_r=ar^3+br^2+cr+d with four unknowns.

  2. Substituting r=1,2,3,4r=1,2,3,4 gives four equations.

  3. Solve the system (calculator).

Answer · Q

Tr=r3/6−r2/2+4r/3T_r=r^3/6-r^2/2+4r/3.

Explore the idea

A cubic through four points predicts T₅ = 15

The curve is T(r) = r³/6 − r²/2 + 4r/3. It passes through the four given terms, but not through 16 at r = 5.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Cubic through the four terms
  • Powers of 2 (not what the question says)
Ideas to try
  1. Compare the two curves at r=5r=5. By how much do they differ?
  2. Where else do they agree?