Arithmetic Sequences - Problem 2
Finding the general term for an arithmetic sequence when we don’t really know the first term or the difference. There’s a couple ways of doing this.
The first way the way I’m going to focus on right now is we are basically just given two numbers right next to each other, we are given the 20th term of the sequence and the 20st term of the sequence. So using those two I can easily find the difference. I went up one term number and I went up three units, so what this tells me is that my common difference is just 3.
I then need to make a equation using the general term to find my first term. So what the general term is is a sub n is equal to a sub 1 plus n minus 1 times d. And we can choose either one of these pieces of information to plug in in order to solve for a sub 1, the piece of information we're missing. I’m going to choose to work with a sub 21 because this is representing n so when I plug the 21 in here 21 minus 1 gives us 20 which is a lot nicer number than if I plug this is 20 minus 1 then I’m dealing with a 19.
You get the exact same answer but just to make our numbers easier I’m going to go with this one. So we know we are dealing with this term and the actual term is 104. So we can plug that in for a sub n a sub 1 is what we don’t know, it’s what I need to find plus n minus 1, the 21 is our nth term so this is just going to be 21 minus 1 which is 20 and times our d which we know to be 3.
We now have enough information to solve for a1, 104 that is equal to a1 plus 21 minus 20 times 3 is 60 so a1 is equal to 104 minus 60 which is just 44.
So now I have enough information to make my general term. I know my first term and I know my d, so I just have to plug those into the equation leaving me with a sub n is equal to a1 which we just found to be 44, plus n minus 1 times d which we know to be 3.
So taking some information, finding our missing link and then putting it together to find our general term.