Linked progressions, progressions in context, trigonometric terms, and sums in a range

Last time

Before we start: last time, in Approximating with an expansion, progressions, and coefficients in progressions, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

The same terms, two rules

Here are four faces of the same topic. One set of terms treated as arithmetic, then as geometric. A real situation where prices climb by a percentage, and the total over the years is a geometric sum. Terms made of sines and cosines, and a sum that starts and finishes part way along the sequence. Every one comes back to nth term and sum formulas.

The four formulas

Arithmetic: the nth term is a plus n minus 1 d. The sum of n terms is n over 2 times 2a plus n minus 1 d. Geometric: the nth term is a r to the n minus 1. The sum of n terms is a times r to the n minus 1, over r minus 1. The sum to infinity, when the ratio is between minus 1 and 1, is a over 1 minus r.

The same terms, two rules

The first three terms of an arithmetic progression are p squared over 6, 2p minus 6 and p. For an arithmetic progression, twice the middle term equals the sum of the outer two. Rearrange: p squared over 6 minus 3p plus 12 is zero. So p squared minus 18p plus 72 is zero, and p is 6 or 12. But p equal to 6 gives 6, 6, 6, so the difference is zero. The question says it is not, so p is 12. Now the first two terms are 24 and 18, a geometric progression with ratio three quarters. The sum to infinity is 24 over a quarter, which is 96.