Moderate -0.8 This is a straightforward application of standard summation formulae from the formula booklet. Students need to split the sums, apply the given formulae for Σr², Σr, Σr³, and perform arithmetic to verify equality. While it requires careful algebraic manipulation across two separate calculations, it involves no problem-solving insight or novel techniques—purely mechanical application of memorized formulae.
1 Use the List of Formulae (MF10) to show that \(\sum _ { r = 1 } ^ { 13 } \left( 3 r ^ { 2 } - 5 r + 1 \right)\) and \(\sum _ { r = 0 } ^ { 9 } \left( r ^ { 3 } - 1 \right)\) have the same numerical value.
1 Use the List of Formulae (MF10) to show that $\sum _ { r = 1 } ^ { 13 } \left( 3 r ^ { 2 } - 5 r + 1 \right)$ and $\sum _ { r = 0 } ^ { 9 } \left( r ^ { 3 } - 1 \right)$ have the same numerical value.
\hfill \mbox{\textit{CAIE FP1 2015 Q1 [4]}}