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Question Number 89768 by john santu last updated on 19/Apr/20

$$t\left(n\right)-t(n-1)=n$$$$forn>0andt\left(0\right)=1$$$$findt\left(n\right)$$

Answered by john santu last updated on 19/Apr/20

Answered by mr W last updated on 19/Apr/20

$$t\left(k\right)-t(k-1)=k$$$$\underset{k=1}{\sum ^n}t\left(k\right)-\underset{k=1}{\sum ^n}t(k-1)=\underset{k=1}{\sum ^n}k$$$$\underset{k=1}{\sum ^n}t\left(k\right)-\underset{k=0}{\sum ^{n-1}}t\left(k\right)=\underset{k=1}{\sum ^n}k$$$$\underset{k=0}{\sum ^{n-1}}t\left(k\right)+t\left(n\right)-t\left(0\right)-\underset{k=0}{\sum ^{n-1}}t\left(k\right)=\underset{k=1}{\sum ^n}k$$$$t\left(n\right)-t\left(0\right)=\underset{k=1}{\sum ^n}k=\frac{n(n+1)}{2}$$$$t\left(n\right)=t\left(0\right)+\frac{n(n+1)}{2}$$$$\Rightarrow t\left(n\right)=1+\frac{n(n+1)}{2}$$

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