Thank you for these solutions to Shahnawaz
Abdullah;  Daniel Disegni, Liceo Scientifico
Copernico, Torino, Italy; Anderthan Hsieh,
Saratoga High School; Andrei Lazanu, School 205,
Bucharest, Romania; David Moxey, Queen Mary's
Grammar School, Walsall; Paddy Snow; Peter
Barton, Greshams School, Holt, Norfolk;  Ngoc
Tran, Nguyen Truong To High School (Vietnam);
Chris Tynan, St. Bees School; Dorothy Winn,
Madras College; A Ji Yang; Hyeyoun Chung, St.
Paul's Girls' School; Yatir Haslevi.
<p/>
To prove that 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>k</mi><mo>.</mo><mi>k</mi><mo>!</mo><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mo>-</mo><mi>k</mi><mo>!</mo></mrow></math>.
<p/>
If we take 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>k</mi><mo>!</mo></mrow></math> out as a factor from the right
hand side of the equation, we are left with

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>k</mi><mo>!</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math> which simplifies to 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>k</mi><mo>.</mo><mi>k</mi><mo>!</mo></mrow></math>, as
required.
<p/>
Now we sum the series 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mn>1</mn><mo>.</mo><mn>1</mn><mo>!</mo><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mi>n</mi><mo>.</mo><mi>n</mi><mo>!</mo></mrow></math>
<p/>
As we have proved, 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi><mo>.</mo><mi>n</mi><mo>!</mo></mrow></math> is equal to 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mo>-</mo><mi>n</mi><mo>!</mo></mrow></math> and therefore 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo><mo>.</mo><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></math> is equal to

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mo>-</mo><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></math> which simplifies to 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi><mo>!</mo><mo>-</mo><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></math>. If we add the two results, we find that

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi><mo>!</mo></mrow></math> cancels.
<p/>
If we sum the series from 1 to 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math>, we find that
all of the terms cancel except for 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></math> and

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo>-</mo><mo stretchy="false">(</mo><mn>1</mn><mo>!</mo><mo stretchy="false">)</mo></mrow></math>. Thus the sum of all numbers of the form

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>r</mi><mo>.</mo><mi>r</mi><mo>!</mo></mrow></math> from 1 to 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math> is equal to 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mo>-</mo><mn>1</mn></mrow></math>.
<p/>

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