Double sigma summation is in complexity calculation

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Basically i was reading skiena and doing exercise of 2nd chapter.The result of my calculation got differed from the actual solution given on Solution site and there is one thing i don't understand how to prove

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Is only method to prove is induction?

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$$\sum_{j=i+1}^n j=(i+1)+\ldots+n=(1+\ldots+i)+[(i+1)+\ldots+n]-(1+\ldots+i)=\sum_{j=1}^n j-\sum_{j=1}^i j$$