Grades 11-12 Video Solutions 2012-2012_11-12

Video Transcription

In the sequence 1, 1, 0, 1, negative 1, and so on, each of the first two terms, a1 and a2, is 1.  The third term is the difference of the two preceding terms, that is, a3 equals a1 minus a2. The fourth is the sum of the two preceding terms, that is, a4 equals a2 plus a3.  Then a5 equals a3 minus a4, and a6 equals a4 plus a5, and so on. What is the sum of the first hundred terms of this sequence?  All right, so let's just write things out to see if we see a pattern. So we start with some a1 and a2. They end up being 1, but at this point it doesn't really matter.  So a3 is a1 minus a2. a4 is a2 plus a3, and now let's substitute that back in to get just a1, because this a2 and this minus a2 cancel.  Now let's figure out a5 is a3 minus a4. Substitute everything in, and we get minus a2. a6 is a4 plus a5. Substitute everything again, we get a1 minus a2.  Then a7, substitute everything in, and we get negative a1. And then a8 ends up being negative a2.  Okay, so whenever we calculate terms, if it's an odd term, we subtract, and if it's an even term, we add.  So it kind of matters whether we're at an odd step or an even step in our process here. So let's see what happens.  We went from a1 to a7, so this goes from 1 is odd and 7 is odd, so it stayed odd, and we just negated. And a2 to a8 also negated, and 2 and 8 are both even.  So this is kind of telling us that every 6 terms in our sequence, we just negate everything,  because if I started with minus a1 and minus a2, all of the successive terms would be the negative of what they currently are.  So that means if I add up a1 and a7, it just cancels out. If I add up a2 and a8, it cancels out.  And that means the sum of the first 12 terms, meaning the first 6 terms, which have one sign,  and the next 6 terms, which have the opposite sign, the sum of the first 12 terms is just 0. So that means every time we have a group of 12 terms, they add up to 0.  So now that we want the sum of the first 100 terms, well, the first 12 times 8, which is 96 terms, just end up adding up to 0, and all that matters is the next 4 terms.  And they've given us these 4 terms right here, 1, 1, 0, and 1, which add up to 3.

Video Summary

In the given sequence, each term is calculated based on the two preceding terms, with alternating signs based on whether the step is odd or even. Every set of six terms is a negation of the previous six terms, causing their sums to cancel out. Hence, every 12 terms add up to zero. For the first 100 terms, the first 96 terms, which include eight full 12-term cycles, sum to zero. The remaining four terms are 1, 1, 0, and 1, summing up to 3. Therefore, the sum of the first 100 terms is 3.

Keywords

sequence

alternating signs

term calculation

cycle cancellation

sum of terms