Sum[Floor[x + k/y], {k, 0, y}] == Floor[x y + (Ceiling[y] - y) Floor[x + 1]] /; Element[x, Reals] && Element[y, Reals] && y > 0

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> y </mi> </munderover> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mi> k </mi> <mi> y </mi> </mfrac> <mo> + </mo> <mi> x </mi> </mrow> <mo> &#8971; </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> &#8968; </mo> <mi> y </mi> <mo> &#8969; </mo> </mrow> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </mrow> <mo> &#8971; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> y </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> y </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> y </ci> </uplimit> <apply> <floor /> <apply> <plus /> <apply> <times /> <ci> k </ci> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> x </ci> </apply> </apply> </apply> <apply> <floor /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ceiling /> <ci> y </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> <apply> <floor /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <in /> <ci> y </ci> <reals /> </apply> <apply> <gt /> <ci> y </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "y_"], RowBox[List["Floor", "[", RowBox[List["x_", "+", FractionBox["k_", "y_"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Floor", "[", RowBox[List[RowBox[List["x", " ", "y"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Ceiling", "[", "y", "]"]], "-", "y"]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List["x", "+", "1"]], "]"]]]]]], "]"]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["y", "\[Element]", "Reals"]], "&&", RowBox[List["y", ">", "0"]]]]]]]]]]

2001-10-29