The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc. The n th partial sum is given by a simple formula: ∑ k = 1 n k = n ( n + 1 ) 2 . {\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}}.}

2017年7月6日 · To find the sum of the numbers from 1 to 100, we can use the formula for the sum of an arithmetic series. The formula is: where S is the sum, n is the number of terms, a is the first term, and l is the last term. In this case, n = 100, a = 1, and l = 100. Plugging these values into the formula, we get:

Every integer n can be written as n= (n+1)+(n+2)-(n+3) where (n+1) and (n+2) are of course two consecutive integers.

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An example computation to help you find a general formula for k\star S_n which may be proven by induction. Observe that 3\star S_n=\sum_{m=1}^n2\star S_{m}=\sum_{m=1}^n\sum_{j=1}^m1\star S_j= \sum_{m=1}^n\sum_{j=1}^m\sum_{i=1}^j\binom{i}{1} ...

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