# Arithmetic and Geometric sequences - Describe the following sequence

So I'm asked to determine the type of sequence below as well as state the $$a$$, $$d$$, and $$r$$ values. I know the answers to a, b, and c, but for the last one I'm confused as to what to categorize it under. I was thinking $$a$$ would be $$65$$, but that's all I could get. I can't figure out the d value, and I don't know whether it's geometric or arithmetic. Any help or hints would be appreciated.

$$y = 77x - 12, x\in\mathbb{Z}^+$$

I couldn't figure out how to format the equation properly, so I included a picture with the question below. I'm struggling with d)

As stated in the comments, a is the base value, d is the difference between each element in an arithmetic sequence, and r is the ratio between members in a geometric sequence. So like if I was given the pattern $$3, 5, 7, \ldots$$

$$a = 3$$, and $$d = 2$$. There is no $$r$$ value because it's an arithmetic sequence.

• I don't believe "the $a,d$ and $r$ values" are standard terminology. Can you define them? – lulu Jun 13 '19 at 20:32
• The sequence $x_n$ is given by putting in consecutive positive integers (? are you sure here? you have a + and the task has a *) for $x$ in your equation. What is the first member of this sequence? What is the second? Did you figure it out? – B.Swan Jun 13 '19 at 20:33
• @lulu My guess is that $a$ is the base value of an arithmetic sequence, $d$ is the difference of each member of the sequence, and $r$ is the ratio of members in a geometric sequence – B.Swan Jun 13 '19 at 20:36
• @B.Swan Sure, that's possible...but the notation is odd. $x^a$ is hard to understand. I guess that $a$ has nothing to do with the other $a$? The picture says $x_n=77x-12$ but I guess they meant to write $x_n=77n-12$? Seems like we're being asked to guess at a whole lot of things. – lulu Jun 13 '19 at 20:40
• Have you tried substituting the first few positive integers for $x$? – N. F. Taussig Jun 13 '19 at 21:25

## 2 Answers

The last equation (part d) means (though ambiguous) that you get the $$n^{th}$$ term of the sequence by plugging the values of $$x$$ in the R.H.S. For example - The first term of the sequence can be obtained by putting $$x=1$$ and it yields $$(77*1-12=65)$$. The $$10^{th}$$ term (for example) can be found by putting $$x=10$$ which yields $$(77*10-12=758)$$.

Now it is given that the domain of $$x$$ is the set of all positive integers i.e. $$1,2,3...$$ . So the first term is $$65$$ (put $$x=1$$). Second term is $$142$$ (put $$x=2$$). Third term is $$219$$ (put $$x=3$$). So it is clear that the sequence is an Arithmetic Progression with common difference $$77$$.

Note - a, r and d are not fixed notations. You can rather say a = first term of the sequence, d = common difference of an Arithmetic Progression, r = $$r^{th}$$ term of the sequence.

COMMENT.- Question $$d$$ seems to me somewhat ambiguous. One way to interpret it is, for example, to take a positive integer arbitrary $$x_0$$ and define the sequence $$x_1=77x_0-12\\x_2=77x_1-12\\x_3=77x_2-12$$ and so on so we have $$x_n=77^nx_0-12(77^{n-1}+77^{n-2}+\cdots+77+1)\\x_n=77x_0-12\frac{77^n-1}{77-1}=\frac{77^n(19x_0-3)+3}{19}$$ where $$19x_0-3=a\in\mathbb N$$ so $$x_n=\frac{77^na+3}{19}$$ However that is an interpretation of a question somewhat ambiguous as I said above.....