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<br>To extend Mitch Wolfe’s contribution:<div><br></div><div>Geometric progressions can indeed blow up. </div><div>In the limit of small increments, the geometric progression turns into the exponential, the function that can be proved to blow up faster than any other function.</div><div><br></div><div> Mike Alexander<br><br><br><br><p class="yahoo-quoted-begin" style="font-size: 15px; color: #715FFA; padding-top: 15px; margin-top: 0">On Wednesday, September 27, 2023, 2:59 PM, Steve Parus via LCTG <lctg@lists.toku.us> wrote:</p><blockquote class="iosymail"><div id="yiv5575682670"><div><div dir="ltr"></div><div dir="ltr">From ‘The Magic of Numbers’ by Robert Tocquet, A. S. Barnes and Company: </div><div dir="ltr"><br clear="none"></div><div dir="ltr">The problem of the Folded Paper page 109</div><div dir="ltr"><br clear="none"></div><div dir="ltr">The String Round the Equator page 112</div><div dir="ltr"><br clear="none"></div><div dir="ltr">Steve Parus</div><div dir="ltr"><div id="yiv5575682670yqtfd68698" class="yiv5575682670yqt8360716009"><br clear="none"><blockquote type="cite">On Sep 27, 2023, at 2:30 PM, Mitchell I. Wolfe via LCTG <lctg@lists.toku.us> wrote:<br clear="none"><br clear="none"></blockquote></div></div><div id="yiv5575682670yqtfd39492" class="yiv5575682670yqt8360716009"><blockquote type="cite"><div dir="ltr"></div></blockquote></div></div><div id="yiv5575682670yqtfd93507" class="yiv5575682670yqt8360716009"></div><div><div id="yiv5575682670yqtfd47408" class="yiv5575682670yqt8360716009"><p>The classic <a rel="nofollow noopener noreferrer" shape="rect" target="_blank" href="https://en.wikipedia.org/wiki/Wheat_and_chessboard_problem">wheat and chessboard problem</a> first recorded in 1256 also exhibits similar characteristics. Sometimes people start with a penny instead of a grain of wheat. Either way the nation's treasury is depleted well before the 64th square.</p>
<p>Even at much lower increments, Albert Einstein famously referred to compound interest as the eight wonder of the world.</p>
<p>Geometric progressions can blow up...</p>
<p>On 2023-09-27 13:40, Jon Dreyer via LCTG wrote:</p>
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<p><span style="font-family:Times New Roman, Times, serif;">2^10 is about 1000, so 2^50 would be about (2^10)^5 which is about 1000^5 or 10^15.</span></p>
<p><span style="font-family:Times New Roman, Times, serif;">A ream of paper is 500 sheets and about 2" high so 1000 sheets would be 4" high so each sheet is 4*10^-3 high. Multiplying that by 10^15, which is the number of thicknesses of paper, we get 4*10^12". To convert to miles, divide by 12*5280, which is a bit more than 5*10^4, so we get somewhat less than 10^8 miles. An astronomical unit (Earth-Sun distance) is around 93,000,000 miles (I can still hear my dad's voice telling me that when I'm still a little kid) or about 10^8. So the stack of paper would approximately reach the Sun, way farther than the moon.</span></p>
<p><span style="font-family:Times New Roman, Times, serif;">Checking my work, the height of the paper would be</span></p>
<p><span style="font-family:Times New Roman, Times, serif;">(2**50)*(4*10.0**(-3))/(12*5280)<br clear="none">=> 71079539.57339798</span></p>
<p><span style="font-family:Times New Roman, Times, serif;">which is indeed the same order of magnitude as an astronomical unit. Of course the mass of that folded stack of paper would be so huge that we'd have crashed into the Sun, or vice versa. So kids, don't try this at home!<br clear="none"></span></p>
<p>--</p>
<p style="font-family:Times, serif;">Jon "Overshoot The Moon" Dreyer<br clear="none"><a rel="nofollow noopener noreferrer" shape="rect" target="_blank" href="http://www.passionatelycurious.com">Math Tutor/Computer Science Tutor</a><br clear="none"><a rel="nofollow noopener noreferrer" shape="rect" target="_blank" href="http://music.jondreyer.com">Jon Dreyer Music</a></p>
<div class="yiv5575682670v1v1moz-cite-prefix">On 9/27/23 1:04 PM, Steve Parus via LCTG wrote:</div>
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<div dir="ltr">The change in diameter is equal to the change in circumference divided by pi. </div>
<div dir="ltr"> </div>
<div dir="ltr">Another one: if a piece of paper is repeatedly folded in half 50 times, would its thickness reach the moon?</div>
<div dir="ltr"> </div>
<div dir="ltr">Steve Parus</div>
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