AP space and difference operator

Problem Statement

Let $\mathcal{A}(a,d)$ denote an AP. Define

• Show any AP is uniquely $a\mathcal{I}+d\mathcal{J}$.
• Let $\Delta x_n=x_{n+1}-x_n$. Show $\Delta\mathcal{I}=0$ and $\Delta\mathcal{J}=\mathcal{I}$.
• Deduce $\Delta(a\mathcal{I}+d\mathcal{J})=d\mathcal{I}$.

Hints

Compare first term and common difference.

Solutions

An AP has $n$th term $a+(n-1)d=a\cdot1+d(n-1)$, so $X=a\mathcal{I}+d\mathcal{J}$, unique by matching $(a,d)$. Also $\mathcal{I}=(1,1,1,\dots)$ so $\Delta\mathcal{I}=0$, and $\mathcal{J}=(0,1,2,\dots)$ so $\Delta\mathcal{J}=(1,1,1,\dots)=\mathcal{I}$. Hence

Further Readings

• HSC Vectors: https://vumaths.com/booklets/hsc-vectors/
• HSC Differential Equations: https://vumaths.com/booklets/hsc-differential-equations/
• HSC Combinatorics: https://vumaths.com/booklets/hsc-combinatorics/
• HSC Mechanics: https://vumaths.com/booklets/hsc-mechanics/

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