ANSWERS: 1

You haven't provided very much information, so I have to go off of some assumptions. I'm assuming you mean that a1 is the first term of a geometric sequence and that a4 is the fourth term of the same. If each term is given as an = ar^n, then the common ratio would be a really messy irrational number, so, I'll assume that an = ar^(n1) instead. It's a bit weird of a way to denote things, but I am fairly confident that you are either not stating the information properly or that your teacher is, for whatever reason, using some nonstandard notation for the sequence terms. At any rate, the three terms of interest are "n", the cardinal number of the term in the sequence, "a" the scale factor, and "r" the common ratio. Since both terms can have 5/3 factored out of them, I'll say that 5/3 is likely the scale factor. 5/3 * r^(11) = 5/3, so a1 works out perfectly, and 5/3 * r^(41) = 5r^3 / 3. If a4 = 5/192, then 5r^3 / 3 = 5/192. Multiply both sides of the equation by 3/5 and r^3 is revealed to be 1/64, so r = 1/4.
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