Let us continue with introducing the two types of sequences. The first one is the arithmetic sequence, defined as a sequence with a constant difference between consecutive terms. The constant difference is represented by the variable d and the first term by the variable a. For instance, the sequence of even numbers is an arithmetic sequence, because all the terms are consecutive with a difference of 2, which is d. Variable a would be 0, the first even number. As you can expect, there are formulas for an arithmetic sequence, recursive and direct. Xi = Xi-1 + d is the recursive formula, as you can see that Xi-1 is the previous term and must be added a certain amount d in order to get to the next term. For direct, the formula is Xi = X1 + (i – 1)d.
Geometric sequence is a sequence with a constant ratio between consecutive terms. In other words, in order to get from one term to the next, you have to multiply or divide by a constant r. One example is 2,4,8,16,32, etc. Here, r is 2 because in order for one term to go on to the next, you have to multiply by 2. This all translates into a geometric sequence’s recursive formula: Xi = r(Xi-1). The direct formula in this case is Xi = a(ri-1), where r is the constant ratio and a is the first term.
Now it’s time to put your sequences knowledge from yesterday and today to the test. Here are a few questions for you to try out:
1) What is the 20th term in the geometric sequence 1/20, 1/10, 1/5, 2/5…?
2) If the 6th term of an arithmetic sequence is 21 and the constant difference is 3, then what is the first term?
3) Finally, there is a geometric sequence an with n= 1,2,3…etc. Assume a1=2 and a3=4. Find the value of a7.
Tomorrow, I will post up just the answers. If you want to know the process, contact me tomorrow after I post up the answers. I will also for tomorrow introduce series and present the proof of the formula of a geometric series. That’s all for today.