Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

In a sequence, the first term, a1, is between 0 and 1. For all n greater than or equal to 1, a sub 2n is equal to a2 times an plus 1, and a2n plus 1 is equal to a2 times an minus 2.  Given that a7 is equal to 2, what is the value of a2?  So first let's start experimenting out with this recurrence that they've given us. Let's write out what a2 is supposed to be by plugging in n equals 1 here.  We get that a2 is equal to a2 times a1 plus 1. Well now if I plug in n equals 1 here, I can calculate a3 to get a3 is a2 a1 minus 2.  Well we already know that a2 is equal to a2 a1 plus 1, and now this a2 a1 is shared. So if I subtract over the 1 and then substitute in a2 minus 1 over here, we get that a3 is  equal to a2 minus 3, which is I think a little bit nicer than this thing over here. Well now how can we calculate a7? Well 7 is odd, so we probably want to use this one right here.  And in order to get 2n plus 1 to be 7, we plug in n equals 3. And we know a3, so that's good. So let's plug in 3 in here to get that a7 is equal to a2 a3 minus 2.  And I'm going to plug in this new expression for a3 right here to get a2 squared minus 3 a2, and then the minus 2 stays the same.  We know that a7 is equal to 2, so I'm just going to set this equal to 2. Now this is a quadratic equation in terms of a2.  If I move over this 2 and factor, it factors as a2 minus 4 times a2 plus 1, meaning we have two options for a2, that it's negative 1 or positive 4. Well now which one is it?  Well we know that the first term a1 is between 0 and 1, so now let's use this right here to solve for a1.  So if I subtract over the 1 and divide by a2, I get that a1 is equal to a2 minus 1 over a2. Plugging in minus 1 for a2 gets me that a1 is 2, and that's outside the range.  Plugging in a2 equals 4 gets me a1 is 3 fourths, which is inside the range. So a2 equals 4.

Video Summary

Summary Not Available