Playing with a Calculator #1: the Sierra Club's 2% solution

Continuing from the previous post: The Sierra’s Club’s 2% solution--reduce your emissions 2% a year for 40 years--seemed too good to be true, and at first I thought there was something hinky with their numbers, especially when I took our first-year number, 270 lbs, and multiplied it by 40, yielding 10,800 lbs: that seemed absurdly easy, but then I realized that only tells us the reduction of the final year. Since we are currently reducing by a lot more than 270 pounds, the number I really wanted is the total number of pounds over the whole period. Using an all-too-human process of rationalization, I figured that if we accelerate our reductions, we will not have to reduce by as much later in the 40-year period.

What Sierra Club doesn't make explicit is that they are using an arithmetic progression. To compute it with round numbers, let’s say 2% for you equals 100 lbs of CO2 emissions. The first year you reduce by 100, the next by 200, then 300, and so on until in year 40, you have to reduce by 4000 lbs. Arithmetic progression has its own cool factoid concerning Carl Friedrich Gauss, whose teacher asked his third-grade class to calculate the sum of the numbers 1-100. The 10 year old child immediately popped out with the answer, 5050.

For those of us who still struggle with calculating 15% tips, the formula is:

N(a1 + an)/2=Sn

Again with our example:

40(100 + (100 x 40)/2=82,000

or a total of 82,000 pounds over the whole period.

Our family’s number was 270 lbs, so our 40-year total is 221,400 lbs. That averages to a 2.76 ton reduction each year—roughly 40%.

The old faithful, Wikipedia, has a helpful explanation of the formula, including the Gauss factoid.