t-n-t-n-1-n-for-n-gt-0-and-t-0-1-find-t-n-

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

t (n) - t (n - 1) = n

f o r n > 0 a n d t (0) = 1

f i n d t (n)

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

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

t (k) - t (k - 1) = k

\underset{k = 1}{\sum^{n}} t (k) - \underset{k = 1}{\sum^{n}} t (k - 1) = \underset{k = 1}{\sum^{n}} k

\underset{k = 1}{\sum^{n}} t (k) - \underset{k = 0}{\sum^{n - 1}} t (k) = \underset{k = 1}{\sum^{n}} k

\underset{k = 0}{\sum^{n - 1}} t (k) + t (n) - t (0) - \underset{k = 0}{\sum^{n - 1}} t (k) = \underset{k = 1}{\sum^{n}} k

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

t (n) = t (0) + \frac{n (n + 1)}{2}

\Rightarrow t (n) = 1 + \frac{n (n + 1)}{2}

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