Question

Give an example of the following: an arithmetic sequence written in explicit form.

Answer 1

What is "explicit form"? Do they mean an explicit formula? Or should I just list out the numbers for an arithmetic sequence (ie. 1, 2, 3, 4, 5, etc.)?

Answer 2

@jim_thompson5910 Do you know?

Answer 3

Formula, I believe. The number sequence is "implicit" since it only "implies" that there is a rule, it doesn't tell you the rule.

Answer 4

So something like: y = 5x would work?

Answer 5

Or y = 5x + 1... the arithmetic "rule" would be add 5 every time... the sequence would be 1, 6, 11, etc.

Answer 6

`Or should I just list out the numbers for an arithmetic sequence (ie. 1, 2, 3, 4, 5, etc.)?`

the terms never end. It's an infinite sequence. So you cannot list out all the terms. You're better off writing the formula for the nth term

Answer 7

No, that doesn't really express a sequence, just a line.

I found this: \[a _{n} = a _{1} + (n – 1)d\]on
http://www.mathwords.com/a/arithmetic_sequence.htm

Answer 8

Is that the "explicit form" though?

Answer 9

Thanks @mjdennis I'll go take a look

Answer 10

\[a _{n+1}=a _{n}+1\]
where n is an integer.

Answer 11

Perfect. I'm assuming "explicit form" is the same as "explicit formula"?

Answer 12

That is a general formula for any explicit formula.
Pick some numbers for a1 and d

Answer 13

For the sequence 1, 6, 11, 16, 21... it would be y = 5(x-1) + 1 correct? thanks for you help.

Answer 14

Yes, except you should really replace "y" with a(n) and "x" with n, but I think you got the concept!

Answer 15

Thanks!

Answer 16

Just in case there was any confusion,
\(\large\rm a_n=a_{n-1}+1\)
is the `recursive formula` that gives the sequence.
Each number relates to the previous number in some way.

\(\large\rm a_n=a_1+(n-1)\)
is the `explicit formula` because you can plug in an n and solve for a_n directly without any recursion :)