Problem 44513. Add all the numbers between two limits (inclusive)

Created by David Verrelli

Solve Later

{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>In this problem you must add up \"all of the numbers\" between two specified limits,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b</w:t></w:r><w:r><w:t>, in which</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> ≤</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b</w:t></w:r><w:r><w:t>. However, the practical interpretation of \"all of the numbers\" will depend upon the specified</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/numeric-types.html\"><w:r><w:t>data type</w:t></w:r></w:hyperlink><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>dt</w:t></w:r><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Mathematically speaking, if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> &lt;</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b</w:t></w:r><w:r><w:t> then the required sum constitutes an</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>infinite series</w:t></w:r><w:r><w:t> that does not converge (i.e. the required sum would be infinity). For example, if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a=1</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b=2</w:t></w:r><w:r><w:t> then we could capture</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>some</w:t></w:r><w:r><w:t> of those numbers through the series lim n→∞ ⁿ∑ᵢ₌₁{1 + (1/i)} = lim n→∞ {n + ⁿ∑ᵢ₌₁(1/i)} ≈ lim n→∞ {n + γ + ln(n)}, using properties of the harmonic series in the last approximation.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>But MATLAB cannot represent numbers with</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>infinite</w:t></w:r><w:r><w:t> precision. In fact, the precision is determined by the specified</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/numeric-types.html\"><w:r><w:t>data type</w:t></w:r></w:hyperlink><w:r><w:t>. For instance, if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>dt = 'single'</w:t></w:r><w:r><w:t>, then with</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a=1</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b=2</w:t></w:r><w:r><w:t> the summation would comprise the series {(1) + (1+1×2⁻²³) + (1+2×2⁻²³) + (1+3×2⁻²³) + ... + (2−2×2⁻²³) + (2−1×2⁻²³) + (2)} = 12582913.5, which is finite.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Another example:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ % INPUT\n a = 10\n b = 12\n dt = 'int16'\n % OUTPUT\n s = 33 % = 10 + 11 +12]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>So please add up all the numbers between two limits (inclusive), subject to the precision indicated by the specified</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/numeric-types.html\"><w:r><w:t>data type</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>NOTE</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:b/><w:i/></w:rPr><w:t>1</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>: Terminal values</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> and</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>b</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> are whole numbers in every case (albeit implicitly defined as of the</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>double</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> data type); they can be positive or negative. However, values -1&lt;x&lt;+1 are never included in the summations.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>NOTE</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:b/><w:i/></w:rPr><w:t>2</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>: All data types specified in the input</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:r><w:rPr><w:i/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>dt</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> shall be</w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/numeric-types.html\"><w:r><w:rPr><w:i/></w:rPr><w:t>numeric</w:t></w:r></w:hyperlink><w:r><w:rPr><w:i/></w:rPr><w:t>.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

In this problem you must add up "all of the numbers" between two specified limits, <tt>a</tt> and <tt>b</tt>, in which <tt>a</tt> ≤ <tt>b</tt>. However, the practical interpretation of "all of the numbers" will depend upon the specified <a href = "https://au.mathworks.com/help/matlab/numeric-types.html">data type</a>, <tt>dt</tt>.Mathematically speaking, if <tt>a</tt> &lt; <tt>b</tt> then the required sum constitutes an infinite series that does not converge (i.e. the required sum would be infinity). For example, if <tt>a=1</tt> and <tt>b=2</tt> then we could capture some of those numbers through the series 
lim n→∞ ⁿ∑ᵢ₌₁{1 + (1/i)} = lim n→∞ {n + ⁿ∑ᵢ₌₁(1/i)} ≈ lim n→∞ {n + γ + ln(n)}, using properties of the harmonic series in the last approximation.But MATLAB cannot represent numbers with infinite precision. In fact, the precision is determined by the specified <a href = "https://au.mathworks.com/help/matlab/numeric-types.html">data type</a>. For instance, if <tt>dt = 'single'</tt>, then with <tt>a=1</tt> and <tt>b=2</tt> the summation would comprise the series {(1) + (1+1×2⁻²³) + (1+2×2⁻²³) + (1+3×2⁻²³) + ... + (2−2×2⁻²³) + (2−1×2⁻²³) + (2)} = 12582913.5, which is finite.Another example:<pre> % INPUT
 a = 10
 b = 12
 dt = 'int16'
 % OUTPUT
 s = 33 % = 10 + 11 +12</pre>So please add up all the numbers between two limits (inclusive), subject to the precision indicated by the specified <a href = "https://au.mathworks.com/help/matlab/numeric-types.html">data type</a>.NOTE 1: Terminal values <tt>a</tt> and <tt>b</tt> are whole numbers in every case (albeit implicitly defined as of the <tt>double</tt> data type); they can be positive or negative. However, values -1&lt;x&lt;+1 are never included in the summations.NOTE 2: All data types specified in the input <tt>dt</tt> shall be <a href = "https://au.mathworks.com/help/matlab/numeric-types.html">numeric</a>.

Solve

Solution Stats

11 Solutions
4 Solvers

Last Solution submitted on May 16, 2020

Last 200 Solutions

Problem Comments

3 Comments

3 Comments

Tim on 6 Feb 2018

For single (and double) the test suite result for the sum from 2 to 3 is smaller than that for 1 to 2--can that be right?

Tim on 6 Feb 2018

I think I commented too soon again--never mind.

David Verrelli on 7 Feb 2018

No worries, Tim. I found it to be an interesting result too :-) . . . —DIV

Problem Recent Solvers4

Problem Tags

Problem 44513. Add all the numbers between two limits (inclusive)

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers4

Suggested Problems

More from this Author31

Problem Tags

Problem 44513. Add all the numbers between two limits (inclusive)

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers4

Suggested Problems

More from this Author31

Problem Tags

Players