If you were asked how many numbers are there from 26 to 89, would you try to recall the formula taught to us in grade school or would you actually count it?
But if you were asked how many numbers are there from 1 to 25, you KNOW that the answer is 25, because we always start to count from 1 and the last number is the total count.
Since we also KNOW that there are 89 numbers from 1 to 89, removing the first 25 will give us the count from 26 to 89, which is 89 – 25 = 64. In effect we transformed counting from 26 to 89 into counting from 1 to 64.
I Googled “how to find the number of terms in an arithmetic sequence” and I found this in a website: “Use the formula tn = a + (n – 1) d to solve for n. Plug in the last term (tn), the first term (a), and the common difference (d). Work through the equation until you’ve solved for n.”
Another formula is easier to use: n = (tn – a)/d + 1.
But why bother to remember formulas when you can just transform the arithmetic sequence into consecutive counting numbers starting from 1
The key to learning Math is in understanding not memorizing formulas.
In our example, I purposely used “easy-to-compute” numbers so that we can focus on the method not the calculation.
First, “by Addition or by Subtraction” convert the first term into the common difference. Here we subtract 33 from 53 so that the first term will be 20, the common difference.
Next, we also subtract 33 from the last term 333 turning it into 300.
Then we divide the “new” first and last terms by the common difference 20, getting a new sequence of counting numbers from 1 to 15.
Many other examples of counting and other problems in arithmetics sequences are discussed in Chapter 20 of our forthcoming book, “30 Master Strategies In Computing”