How do I calculate the 100th term of the sequence 15, 8, 1, -6...

A sequence means a pattern, so the first thing to do is find the pattern. First try seeing what the difference between the numbers is. 15-7=8, 8-7=1, 1-7=6, so the pattern must be to subtract 7 each time. The first term of the sequence is 15, so let's call that term u₁. The next term, u₂, is u₁-7. The next term, u₃, is u₂-7 or (u₁-7)-7. Let's continue even further, u₄=u₃-7 or (u₁-7-7)-7. Noticing a pattern? u₄ is actually u₁ minus 3*7. Since 3=4-1, we can see that for u₆, we could express it as u₁-(6-1)7, or u₁-57. So we can formulate the general rule u_n= u₁- (n-1)7, where "n" is any number we choose. This question is asking us to find the 100th term. For the 100th term, n=100. So what is u₁₀₀? It's u₁₀₀=u₁- (100-1)7, or u₁₀₀=u₁- (997). Since we know u₁=15, since this is the first term, and 997= 693, we know that u₁₀₀= 15-693= -678.