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 #10
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The following data sets shows first 12 months and the last 12 months of a hypothetical account of 360 interest cycles.

 

The first column displays the $200 deposit; the second column displays the end of month interest payment on the previous balance and the $200 first day of month deposit.

Column 3 displays the new balance.

Column 4 is the multiplier that weights the deposit for the purpose of interest calculation. The multiplier is from 1 to 0, representing the average daily deposit of the $200, where 1 is a deposit on the first day and 0 is a deposit on the last day. (No withdraws are made from this account and only the deposit is weighted).

Column 5 is the average daily of the deposit.

Column 6 is the interest.

Column 7 is the balance.

 

Note that the final balance in column 3 matches the annuity due, while the final balance in column 7 matches the annuity ordinary.

 

 

$$\displaystyle

\noindent \small {First data set. Months 1-12}\\

\begin{tabular}{lllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline

200.00 & 0.50 & 200.50 & 0.00 & 0.00 & 0.00 & 200.00 \\
200.00 & 1.00 & 401.50 & 0.00 & 0.00 & 0.50 & 400.50 \\
200.00 & 1.50 & 603.01 & 0.00 & 0.00 & 1.00 & 601.50 \\
200.00 & 2.01 & 805.01 & 0.00 & 0.00 & 1.50 & 803.01 \\
200.00 & 2.51 & 1,007.53 & 0.00 & 0.00 & 2.01 & 1,005.01 \\
200.00 & 3.02 & 1,210.54 & 0.00 & 0.00 & 2.51 & 1,207.53 \\
200.00 & 3.53 & 1,414.07 & 0.00 & 0.00 & 3.02 & 1,410.54 \\
200.00 & 4.04 & 1,618.11 & 0.00 & 0.00 & 3.53 & 1,614.07 \\
200.00 & 4.55 & 1,822.65 & 0.00 & 0.00 & 4.04 & 1,818.11 \\
200.00 & 5.06 & 2,027.71 & 0.00 & 0.00 & 4.55 & 2,022.65 \\
200.00 & 5.57 & 2,233.28 & 0.00 & 0.00 & 5.06 & 2,227.71 \\
200.00 & 6.08 & 2,439.36 & 0.00 & 0.00 & 5.57 & 2,433.28
\end{tabular}$$

 

$$\displaystyle

\noindent \small {First data set. Months 349-360}\\

\begin{tabular}{lllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
200.00 & 278.06 & 111500.59 & 0.00 & 0.00 & 276.86 & 111222.53 \\
200.00 & 279.25 & 111979.84 & 0.00 & 0.00 & 278.06 & 111700.59 \\
200.00 & 280.45 & 112460.29 & 0.00 & 0.00 & 279.25 & 112179.84 \\
200.00 & 281.65 & 112941.94 & 0.00 & 0.00 & 280.45 & 112660.29 \\
200.00 & 282.85 & 113424.79 & 0.00 & 0.00 & 281.65 & 113141.94 \\
200.00 & 284.06 & 113908.85 & 0.00 & 0.00 & 282.85 & 113624.79 \\
200.00 & 285.27 & 114394.13 & 0.00 & 0.00 & 284.06 & 114108.85 \\
200.00 & 286.49 & 114880.61 & 0.00 & 0.00 & 285.27 & 114594.13 \\
200.00 & 287.70 & 115368.31 & 0.00 & 0.00 & 286.49 & 115080.61 \\
200.00 & 288.92 & 115857.23 & 0.00 & 0.00 & 287.70 & 115568.31 \\
200.00 & 290.14 & 116347.38 & 0.00 & 0.00 & 288.92 & 116057.23 \\
200.00 & 291.37 & 116838.75 & 0.00 & 0.00 & 290.14 & 116547.38
\end{tabular}$$

 

The second data sets are the same as the first, except the multiplier is set to 0.75. This corresponds to a depositor making two deposits of $100 each on the first and 15 of each 30-day month.

 

$$\displaystyle

\noindent \small {Sceond data set. Months 1-12}\\

\begin{tabular}{lllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
200.00 & 0.50 & 200.50 & 0.75 & 150.00 & 0.38 & 200.38 \\
200.00 & 1.00 & 401.50 & 0.75 & 150.00 & 0.88 & 401.25 \\
200.00 & 1.50 & 603.01 & 0.75 & 150.00 & 1.38 & 602.63 \\
200.00 & 2.01 & 805.01 & 0.75 & 150.00 & 1.88 & 804.51 \\
200.00 & 2.51 & 1007.53 & 0.75 & 150.00 & 2.39 & 1006.90 \\
200.00 & 3.02 & 1210.54 & 0.75 & 150.00 & 2.89 & 1209.79 \\
200.00 & 3.53 & 1414.07 & 0.75 & 150.00 & 3.40 & 1413.19 \\
200.00 & 4.04 & 1618.11 & 0.75 & 150.00 & 3.91 & 1617.10 \\
200.00 & 4.55 & 1822.65 & 0.75 & 150.00 & 4.42 & 1821.51 \\
200.00 & 5.06 & 2027.71 & 0.75 & 150.00 & 4.93 & 2026.44 \\
200.00 & 5.57 & 2233.28 & 0.75 & 150.00 & 5.44 & 2231.88 \\
200.00 & 6.08 & 2439.36 & 0.75 & 150.00 & 5.95 & 2437.84
\end{tabular}$$

 

$$\displaystyle

\noindent \small {Sceond data set. Months 349-360}\\

\begin{tabular}{lllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
200.00 & 278.06 & 111500.59 & 0.75 & 150.00 & 277.76 & 111431.07 \\
200.00 & 279.25 & 111979.84 & 0.75 & 150.00 & 278.95 & 111910.02 \\
200.00 & 280.45 & 112460.29 & 0.75 & 150.00 & 280.15 & 112390.17 \\
200.00 & 281.65 & 112941.94 & 0.75 & 150.00 & 281.35 & 112871.52 \\
200.00 & 282.85 & 113424.79 & 0.75 & 150.00 & 282.55 & 113354.08 \\
200.00 & 284.06 & 113908.85 & 0.75 & 150.00 & 283.76 & 113837.84 \\
200.00 & 285.27 & 114394.13 & 0.75 & 150.00 & 284.97 & 114322.81 \\
200.00 & 286.49 & 114880.61 & 0.75 & 150.00 & 286.18 & 114808.99 \\
200.00 & 287.70 & 115368.31 & 0.75 & 150.00 & 287.40 & 115296.39 \\
200.00 & 288.92 & 115857.23 & 0.75 & 150.00 & 288.62 & 115785.00 \\
200.00 & 290.14 & 116347.38 & 0.75 & 150.00 & 289.84 & 116274.84 \\
200.00 & 291.37 & 116838.75 & 0.75 & 150.00 & 291.06 & 116765.90
\end{tabular}$$

 

...

Aug 2, 2015
 #30
avatar+1036 
0

I composed this shortly after MG made his comment. I waited for the post to cool to avoid influence from measurement and comment. (I then forgot about it, until now).

 

All of this over " 256 over 8 plus 15 "

... Wow.

This forum is amazing.

 

You are right MG this forum is amazing.  What you see here is a classic example of social chaos.

 

Generally known as “Chaos Theory,” the chaos in this case is not as much the dictionary definition of “disorder and confusion,” but the mathematical and physics related definition of behavior that is unpredictable and appears as random. It is not truly random though. While this phenomenon cannot be calculated with precision, it can usually be estimated within a range of probable outcomes.

 

Consider the original math question and subsequent answer. It is not likely that anyone could predict the intense social posts that would follow. Though other similar questions have produced similar social posting, there is nothing obvious about why that happens.

 

By the third post, there may be a notable “common denominator” to other posts exhibiting social chaos. TR posted on it. This now begins to explain some of it.When TR posts a reply the probability for chaos increases because TR has a large and verbose social following. While anyone posting a social comment may induce more social comments, TR has a unique gift of turning it into social chaos.

 

A related component of chaos theory is “Catastrophe Theory.” The models for Catastrophe demonstrate how sudden and large changes in behavior arise from small changes in circumstances. Here, again, TR has the unique gift of turning posts into social Catastrophes.

 

After Melody responds to TR’s post and point swiping, the probability the post would continue in the social mode increases greatly. That TR was involved made it a near certainty that social chaos would follow.

 

My name was mentioned which compelled a response from me. I considered using this line to introduce the post: “Hey, I’m your humble and lovable neighborhood troll. . . . .” More Underdog than Spiderman, here.

 

 I didn’t do this because the risk of Melody hurting herself from falling out of her chair following a fit of uncontrollable laughter. (I’m also a considerate troll).

 

Consider the sensitivity to small changes in conditions. For example, what if I wrote, “TR’s military command is similar to the boss p*g on a farm. . . .”? With this, he would retaliate and that would annoy the mods. The is nearly a certainty, because derogatory words and terms are used to enflame the social structure. Some words are so inflammatory they tend lose any useful meaning in a rational conversation.

 

However, If I said, “TR is a skilled student of mathematics,” this would be worse. Again, the mods, and many other forum members, might hurt themselves from excessive laughter. I have a responsibility to limit my comments to prevent potential harm. Owing to the great sensitivity to small changes in conditions, especially coming from a troll, it’s too risky to say those things no matter how much laughter it may bring.

 

On the surface, malicious behavior seems to suffers little consequence for a snarky post that creates an uncontrollable form of social chaos, where things that were once stable begin to spiral out of control, either from rage or laughter. Or both at the same time.

 

I'm a socially responsible troll. I try not to do those things. I usually fail! 

Jul 13, 2015