Tutoring high school math, you encounter arithmetic sequences.  The tutor shows two examples connected with them.

An example of an arithmetic sequence is

10, 14, 18, 22, 26, 30, 34….

You add the same amount each time to get the next term.  In the case above, of course, you add 4 each time.  We call 4 the common difference, while t₁ is 10. t₅ is 26.

Example 1: In the sequence above, what is t₁₀₁?

Solution: We use the formula t_n=t₁+d(n-1)

n is the term number (or the index)
d is the common difference
t₁ is the first term

We plug in the values we know:

t₁₀₁=10+4(101-1)

Now, we simplify the right side. Remember BEDMAS, the order of operations: we must first simplify inside the brackets, then multiply, and finally add:

t₁₀₁=10 + 4(100)=10+400=410

Apparently, t₁₀₁=410.

Example 2: In the above sequence, which term is 202?

Solution: Again, we start with t_n=t₁+d(n-1).

This time, we know t_n is 202; we just don’t know what n is. In other words, we need to know where 202 is in the sequence: Is it the 30th term? The 61st? We need to find out.

We plug in the known values:

202=10+4(n-1)

Next, we subtract 10 from both sides:

192=4(n-1)

We divide both sides by 4:

192/4=n-1

We simplify:

48=n-1

Now, we add 1 to both sides:

49=n

Apparently, n=49 when t_n=202: t₄₉=202. Therefore, in the arithmetic sequence 10,14,18,22,26,30,34…, the 49th term is 202. Or, the index of 202 is 49.

Arithmetic sequences are not always intuitive, but sticking with the formulas and suggested methods can make the questions surprisingly straightforward. I’ll be talking more about them in future posts:)

Source:

Travers, Kenneth et al. Using Advanced Algebra. Toronto: Doubleday Canada Limited,    1977.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.