Four terms identify a cubic · source reference

Printed question

The first four terms of a sequence are given by $T_1=1,T_2=2,T_3=4,T_4=8$. Given that $T_r$ is a cubic polynomial in $r$, find $T_r$ in terms of $r$. [4]

[2011/PJC/Prelim II/1]

Printed working

Let the sequence $T_r=ar^3+br^2+cr+d$.

$$a+b+c+d=1,$$ $$8a+4b+2c+d=2,$$ $$27a+9b+3c+d=4,$$ $$64a+16b+4c+d=8.$$

Using GC: $a=\frac16,b=-\frac12,c=\frac43,d=0$.

$$T_r=\frac16r^3-\frac12r^2+\frac43r.$$

Printed answer key

$T_r=\frac16r^3-\frac12r^2+\frac43r$.

Source provenance

2025 TJC H2 Math (9758) JC2 MSM: EQUATIONS. Question sheet printed pages 15–19; solution sheet printed pages 1–12. Original exam attribution is retained above. Currency dollar signs are spelled out outside maths to avoid confusing currency with TeX delimiters.