Question 1

I don't quite understand the purpose of the recursive formula. I understand how it works, and according to my understanding, in order to find the nth term of a sequence using the recursive definition, you must extend the terms of the sequence one by one. But doesn't this defeat the purpose of it? Isn't the purpose of a formula to find out the nth term of the sequence without computing all the terms before it?

Am I missing something critical here?

Am I missing something critical here?

Accepted Answer

Formulas are just different ways to describe sequences. Each description emphasizes a different aspect of the sequence, which may or may not be useful in different contexts. For example, we may be comparing two arithmetic sequences to see which one grows faster, not really caring about the actual terms of the sequences. In this case, the recursive definition gives the rate of change a little more directly than the standard formula.

There are also sequences that are much easier to describe recursively than with a direct formula. For example, the Fibonacci sequence, which starts {0, 1, 1, 2, 3, 5, 8...}, with each successive term being the sum of the previous two. While this does have a closed formula, it's very complex and unwieldy.

Question 2

Do we have to find the term number before the other ones to find a certain term number?

Accepted Answer

Yes, when using the recursive form we have to find the value of the previous term before we find the value of the term we want to find. For example, if we want to find the value of term 4 we must find the value of term 3 and 2. We are already given the value of the first term.

In other words to find any term beyond the first term we have to start at the beginning which would be the 2nd term and continue to calculate the value of each proceeding term until we have reached the term we want to find.

Makes sense?

Question 3

What good would this stuff do us in the real world? PLZ tell me!

Accepted Answer

Sequences are really important in real life, as they play a key part in areas such as statistics, finance and even in controlling the growth of a species!! One example can be you planning for a vacation. You would look at the temperature of your choosen vacation spot for each month and then decide which month is the apt time to visit the place. Invariably, these temperatures are a sequence and are stored in a set. Who would have known that to enjoy your vacation, you would have to brush up on your sequences first!!

Question 4

Hi. I don't understand what "common difference" stands for.

Accepted Answer

For an arithmetic sequence, we add a number to each term to get the next term. That number is the common difference. 
So for {0, 3, 6, 9...}, we're adding 3 each time. So the common difference is 3.

Note: _only_ arithmetic sequences have a common difference.

Question 5

if the sequence is 4,8,12,16... and arithmetic how could I write a recessive and explicit formula for that sequence?

Accepted Answer

The recursive formula for the arithmetic set{4,8,12,16,...} is: {a(n) = 4 when n = 1
                        a(n-1) + 4 when n > 1
The explicit formula for the same set is: a(n) = 4 + 4(n-1). I hope this makes sense. Thank you.

Question 6

When ever we are doing recursive formulas why do we add that x(n-1)+ something, why do we do that

Accepted Answer

That would be the rule to get any term from its previous term
 
For example,
`c(1)=5`
in order to find any term, we simply need to put the nth term into
`c(n)=c(n−1)+3`
​where +3 is the common difference

Only arithmetic sequences have a common difference
The common difference of an A.P. can be positive, negative or zero

Question 7

What does the *d* mean in  f(n) = f(n − 1) +  *d* ?

Accepted Answer

The "d" represents the common difference (i.e., how much you add/subtract to get the next term in the arithmetic sequence).

Question 8

Can someone explain in #2, how it works/

Accepted Answer

This is the way *I* understand it. First some basics. Since 12 is the starting number you have  d(1)=12. d is  basically saying this is a arithmetic sequence. (1) is saying this is the first number in the sequence and = 12 is saying that that number is 12. Now that that's out of the way, on to the more difficult stuff. You can look at the sequence and see a pattern. What pattern does 12,7,2,-3,-8,... have, well you probably already see that  as each new number is added it is 5 less than the one before it. How would we write that ? Well d(n−1) basically means the number from the number before it's finished product. So like 
d(1)=12 then (d(n-1)-5) = (12-5). Its really simple when you think of it this way. I hope this helped If not tell me so I can try to explain better. : )

Question 9

I'm still confused on why people use recursive formulas. I know they give us the first term and the pattern for a sequence, but don't explicit formulas give us the same information, but without the need for the previous term? Is there any information that recursive formulas do that explicit formulas don't?

Accepted Answer

For some the recursive form is much easier to write and use.  for example a_1 = 1, a_2 = 1 a_n= a_(n-1) + a_(n-2)

Question 10

How would you solve something like:
f(n)=f(n-1)+f(n-1)-f(n-2)+35
f(1)=5
f(2)=30
f(n)=  Some number in the 10thousands, not sure what numbers work in this particular scenario
For n
Is there any way to solve this without going through each and every step?

Accepted Answer

Well, lets see what the first few terms are, f(1) = 5, f(2) = 30, f(3) = 30+30-5+35= 90, f(4) = 90 + 90 - 30+35 = 185, f(5) = 185 + 185 - 90 + 35 = 315, f(6) = 315 + 315 - 185 + 35 = 480.  So we have a sequence of 5, 30, 90, 185,315, 480 ...  We then can find the first difference (linear) which does not converge to a common number (30-5 = 25, 90-30=60, 185-90=95, 315-185=130, 480-315=165.  Then the second difference (60 - 25 = 35, 95-60 = 35, 130-95=35, 165-130 = 35) gives a second common difference, so we know that it is quadratic.  I do not know any good way to find out what the quadratic might be without doing a quadratic regression in the calculator, in the TI series, this is known as STAT, so plugging the original numbers in, I ended with the equation:
f(x) = 17.5x^2 - 27.5x + 15.  This gives us any number we want in the series.  Maybe these having two levels of numbers to calculate the current number would imply that it would be some kind of quadratic function just as if I only had 1 level, it would be linear which is easier to calculate by hand.

$a (n)$ ‍	$= a (n - 1) + 2$ ‍
$a (1)$ ‍			$= 3$ ‍
$a (2)$ ‍	$= a (1) + 2$ ‍	$= 3 + 2$ ‍	$= 5$ ‍
$a (3)$ ‍	$= a (2) + 2$ ‍	$= 5 + 2$ ‍	$= 7$ ‍
$a (4)$ ‍	$= a (3) + 2$ ‍	$= 7 + 2$ ‍	$= 9$ ‍
$a (5)$ ‍	$= a (4) + 2$ ‍	$= 9 + 2$ ‍	$= 11$ ‍

Algebra 1

Course: Algebra 1 > Unit 9

Recursive formulas for arithmetic sequences

How recursive formulas work

Check your understanding

Writing recursive formulas

Check your understanding

Reflection question

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