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Sum of n Terms

If there be ...

Question

If there be $m$ A.Ps beginning with unity whose common difference is $1, 2, 3 . . m .$ Then the sum of their $n^{t h}$ terms is

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Solution

$T_{n}^{1} = (1 + (n - 1) 1)$
$T_{n}^{2} = (1 + (n - 1) 2)$

$T_{n}^{m} = (1 + (n - 1) m)$
Therefore $S_{n}^{m} = n \sum r = 1 (1 + (n - 1) r)$
$= m + (n - 1) \frac{m (m + 1)}{2}$
$= \frac{m}{2} (m n - m + n + 1)$

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