t_1, t_2, t_3, ldots, t_{n} are positive inte | vedclass

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(A) Observe the pattern of the sums: $S_1 = 1^2 = 1$ $S_2 = 3^2 = 9$ $S_3 = 6^2 = 36$ $S_4 = 10^2 = 100$ $S_5 = 15^2 = 225$ The bases of the squares a

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$55$

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(A) Observe the pattern of the sums: $S_1 = 1^2 = 1$ $S_2 = 3^2 = 9$ $S_3 = 6^2 = 36$ $S_4 = 10^2 = 100$ $S_5 = 15^2 = 225$ The bases of the squares are $1, 3, 6, 10, 15, \ldots$. These are the triangular numbers,given by the formula $T_n = \frac{n(n+1)}{2}$. For $n = 10$,the base $k$ is the $10^{th}$ triangular number: $k = T_{10} = \frac{10(10+1)}{2} = \frac{10 	imes 11}{2} = 55$. Thus,$S_{10} = 55^2$,so $k = 55$.

$t_1, t_2, t_3, \ldots, t_{n}$ are positive integers,$S_{n} = t_1 + t_2 + t_3 + \ldots + t_{n}$. Given $S_1 = 1^2, S_2 = 3^2, S_3 = 6^2, S_4 = 10^2, S_5 = 15^2$. Following this pattern,if $S_{10} = k^2$,then $k =$

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