The Great Novena Begins! (and a little piece of math news)

Today we begin the Great Novena to the Holy Spirit, the nine days of that most wonderful countdown to Pentecost. Please pray ferently, we need God's help so much!

Also, I was cranking over a curious recurrence function (more on that another day), and found that I required to know the closed form of the SUM of 3^i (that is, 3-to-the-i) for i running from 0 to n. No doubt you already know the formula, or have a handy book to find it, but I was dumb, and had to work it out for myself.

That is, I wanted T(n) = 1 + 3 + 9 + 27 + ... + 3^n.

Just to be clear, I will write out the first few results:

For n=0 T(0) = 1
For n=1 T(1) = 1 + 3 = 4
For n=2 T(2) = 1 + 3 + 9 = 13
For n=3 T(3) = 1 + 3 + 9 + 27 = 40
For n=4 T(4) = 1 + 3 + 9 + 27 + 81 = 121
For n=5 T(5) = 1 + 3 + 9 + 27 + 81 + 243 = 364

All right... it's obvious, right? The closed form is:
T(n) = (3^(n+1)–1)/2.

The proof is quite easy, and I will leave it to the reader. If you want to see it, let me know.

It reminds me of that hilarious thing we heard long ago, when I was in grad school, about a doll that squawked "Math is Hard!" and how I found out that Aquinas showed the CONTRARY, that Math is actually EASY. Oh yes. But that refers to the DISCIPLINE IN ITSELF, not to how one may find it at any given moment, especially when one has teachers who detest the subect. What a shame. Mathematics does have hard parts, but it can be a lot of fun, and of course every time we say its name we ought to recall our Lord's final words, "Go therefore and MAKE DISCIPLES OF all nations..." (Matthew 28:18) The Greek of course is πορευθέντες οὖν μαθητεύσατε πάντα τὰ ἔθνη, and there we see the root of MATHEMATICS (= The Learning): yes, to learn is to be a disciple. Let us keep that in mind during the Great Novena!

Lord Holy Spirit, enlighten us, in mathematics, and in all our studies, in all our work, in all we do... Amen.