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If we know how to add up the terms of an arithmetic sequence, we could find a closed formula for a sequence whose differences are the terms of that arithmetic sequence. Consider how we could find the sum of the first 100 positive integers (that is, \(T_{100}\)). Instead of adding them in order, we regroup and add \(1+100 = 101\text{.}\) The next pair to combine is \(2+99 = 101\text{.}\) Then \(3+98 = 101\text{.}\) Keep going. This gives 50 pairs which each add up to \(101\text{,}\) so \(T_{100} = 101\cdot 50 = 5050\text{.}\)¹