Little math problem

For pi day here is a little mathematical proof problem for math geeks.

For every natural number n, 20 + 21 + ...+ 2n = 2n+1 − 1.

It's true for n = 1 to 5 as follows:

n=1: 20 + 21 = 3 = 22 − 1

n=2: 20 + 21 + 22 = 7 = 23 − 1

n=3: 20 + 21 + 22 + 23 = 15 = 24 − 1

n=4: 20 + 21 + 22 + 23 + 24 = 31 = 25 − 1

n=5: 20 + 21 + 22 + 23 + 24 + 25 = 63 = 26 − 1

The general formula for all n is proven by mathematical induction as follows and shown here.

Base case: Set n = 0. Then 2⁰ = 1 = 2¹ − 1.

Induction step: Show that if the equation holds for any particular n, which the above does, 
it also holds for n+1.
Let n be any natural number and 2⁰ + 2¹ + ... + 2ⁿ = 2ⁿ⁺¹ − 1 is true.

Then 2⁰ + 2¹ + ...+ 2ⁿ + 2ⁿ⁺¹ = ( 2⁰ + 2¹ + ... + 2ⁿ ) + 2ⁿ⁺¹

                                               = ( 2ⁿ⁺¹ - 1) + 2ⁿ⁺¹

                                               = 2 * 2ⁿ⁺¹ - 1

                                               = 2ⁿ⁺² - 1. QED

There is another proof as well. I will post it this weekend to give readers some time to try to find it on their own.

+++++++++++

March 16. The other proof follows. 2⁰ + 2¹ + ...+ 2ⁿ is a sum of a geometric progression. Using long division, the quotient of (xn+1 − 1)÷(x - 1) is xn + xn-1 + xn-2 +... 1 = xn + xn-1 + xn-2 +...+ x0.
Reverse the order of the sum and let x = 2:
20 + 21 + ... + 2n = (2n+1 − 1)÷(2 - 1) = 2n+1 − 1.

Cost of Socialist Healthcare

Pied pipers of socialism such as Rep. Alexandria Ocasio-Cortez and Sen. Bernie Sanders have high praise for Denmark's and Sweden's healthcare systems, while sweeping the cost, government controls, and other negatives under the rug. The government of Denmark's and Sweden's neighboring country Finland, also with heavy-handed socialist healthcare, has collapsed in large part due to it. I much agree with this Zero Hedge article about Finland.

Included: "The Kaiser Family  Foundation found that 58 percent of Americans oppose “Medicare for all” if told it would eliminate private health insurance plans, and 60 percent oppose it if it requires higher taxes, according to a report by the Washington Free Beacon."

Like I asked five days ago, how much higher taxes? Double all taxes -- income, payroll, and sales -- on yourself and all your friends, relatives, and co-workers who pay taxes?

Amazon HQ2 #4

Cleveland did not make the list of Amazon's top 20 cities contending for its HQ2 announced several months ago. This article, also in yesterday's Cleveland Plain-Dealer print addition, reports that Cleveland, Cuyahoga County, and Ohio offered Amazon about $3.5 billion in incentives to put HQ2 in Cleveland. This wasn't as much offered to locate HQ2 in Alexandria, VA or Queens, NY.

What was offered to locate in Cleveland? There were the typical reduced real estate taxes for many years. In addition, the county subsidy included the creation of a power micro-grid,  which would allow Amazon to power its facilities independent of the main power grid. The micro-grid would cost $200 million, and be paid for by the sale of bonds. Cleveland is believed to have been the only city to offer Amazon an independent and redundant power supply.

Of course, there were other reasons for Amazon not choosing Cleveland -- airport capacity and being an area with a smaller tech workforce than other cities.

Tax Rates for Socialism

Pied pipers of socialism Rep. Alexandria Ocasio-Cortez and Sen. Bernie Sanders are very popular, and the media's attention on them is very high. They love telling voters about how much the voters will receive if the two's socialist proposals are adopted. Free this. Free that. Tax employers and individuals in the top 1% or 10% of income to pay for at all. They ignore or downplay what they propose will cost, even to those who are in the lower 99% or 90% of income. They posit Denmark or Sweden as ideal. They revel in half-truths that are worse than lies. What they don't say is high taxes for all, Venezuela, and Cuba exemplify socialism, too.

Here is a good article that counters their half-truths. Quote: "To turn most self-described socialists into capitalists you really only need to ask one question; would you yourself be willing to pay more?"

How much more? I believe the question should be more realistic, as follows. Would you yourself be willing to pay twice the taxes you do now? That's double all taxes -- income, payroll, and sales. Also ask, Do you want to also double all tax rates on all your friends, relatives, and co-workers who pay taxes? That would raise taxes to about their magnitude in Denmark or Sweden. See the graph on Wikipedia here.

You may note Sweden's 0% income tax rate on the first 18,800 kronor. At the current exchange rate 18,800 kronor is the equivalent of about $2,000 in U.S. dollars. So the average folks pay a 32% income tax rate on most of their income.

Jerry Merryman RIP

Jerry Merryman, co-inventor of the first handheld electronic calculator at Texas Instruments, died February 27 (link). I had not heard of him. I recently read Walter Isaacson's The Innovators. It included a half page about the invention of said calculator. It referred to Patrick Haggerty and Jack Philby, but did not mention Merryman. So I sought to learn a little more about him.

This led me to a two-part article in ElectronicDesign about Merryman and the invention of said calculator. Part 1 download page. Part 2 download page. Registration is free. "How the Computer Got Into Your Pocket" is another story about the early years of the handheld calculator, with more details about Merryman's role.

Income Taxes 2018 #2

One week ago I posted about the many news stories I saw about tax refunds being smaller than last year. By innuendo said stories alleged the Tax Cuts and Jobs Act didn't reduce taxes after all.

Such allegations were based on the earliest filed tax returns. Since then with more returns filed, the average refund has risen to be slightly greater than last year. The Washington Examiner notes that the newsrooms that rushed to report tax refunds were smaller this year -- such as The New York Times, Washington Post, and National Public Radio -- have been silent on the IRS data showing the average refund has since increased.

Income Taxes 2018

I have seen many articles about people receiving lower tax refunds. There is more than one reason why. For some people, such as those living in high tax states, it could be the new limits on itemized deductions. The deductions for state and local taxes, which includes real estate taxes, are now limited to $10,000 due to the Tax Cuts and Jobs Act of 2017. The most common reason, however, is probably the following. New withholding tables quickly followed the Act's passage. For the most part, employers were instructed to withhold less tax from workers in 2018, so that more money would land in their paychecks. As such, much of the money that many filers are expecting this season has already been paid to them via lower withholding last year.

Incidentally, one reason for my lack of posting in the past several weeks is income tax work. Last month I did five days of training in preparation for being an AARP Tax-Aid volunteer three days per week during the tax season, February 1 through April 15. This is my fourth season as a volunteer.