# Nth term of a sequence

First of, I know this is quite easy but I can't really work it out.

I need find the rule for these sequences

$a.$ $2, 3, 4.5, 6.25...$

$b$ $54, 18, 6, 2...$

$c$ $0.01, 0.1, 1, 10...$

My Steps

What I did first was find the differences between the numbers.

For question $a$ the difference between numbers goes $1$, $1.5$, $1.75$

For question $b$ the differnce between numbers goes $-36$, $-12$, $-4$. In this sequence I have found out that the number is being divided by $3$.

For question $c$ the difference between numbers goes $+0.09$, $+0.9$, $+9$. In this the term $9$ is being multiplied by $10$.

I am not sure what to do after. I can do the $nth$ term when it comes to numbers that go up the same and also squared, cubed and triangle numbers.

I know the general formula for $nth$ term is:

nth term = difference x n + (first term - difference).

I can't really use it though as the difference varies. Thanks!

• take ratios of each term to the previous Apr 8, 2014 at 13:09
• For $a$ note that in the sequence of differences, the distance to $2$ halves in each step. Apr 8, 2014 at 13:09
• Infinite answers exist. Suppose a has values corresponding to a 3rd degree polynomial $ax^3+bx^2+cx+d$ at 1,2,3,4 Four equations four variables. Also, you can take any degree polynomial you wish and count at whatever you want... There are infinite answers. Apr 8, 2014 at 13:10
• @Awesome You have to ignore this in order to meaningfully answer these questions. (Try to supply less "entropy" in your answer than given in the sequence information.) Apr 8, 2014 at 13:11
• @PerfectNutter A geometric series takes the form $ar^{n-1}$, where $a$ is the first term and $r$ is the ratio between successive terms. (Be sure to test your guess by plugging in $n=1,2,3,4$ to make sure they match!) Apr 8, 2014 at 13:17

Instead of differences, try the quotients between consecutive terms.

a: Note that the difference of differences is halved at each step.

b: You already answered that - no need to look at the differences, you already stated a rule which generates the $(n+1)$-th term from the $n$-th term. You just need to turn this rule into a formula for the $n$-term.

c: Don't look at the differences here, that only obfuscates things. Just look at how the $(n+1)$-th term is generated from the $n$-th term. It's quite obvious.

• Regarding $a$, you only have two data points for the difference of differences, so you're running low on entropy... Apr 8, 2014 at 13:14

$a.$ $2, 3, 4.5, 6.75...$

$a = 2$ $r = 1.5$ because $3÷2 = 1.5$, $4.5 ÷ 3 = 1.5$ and so on.

The $nth$ term rule is $54 × 1.5^n-1$.

$b.$ $54, 18, 6, 2...$

$a = 54$ $r = $$\frac{1}{3} because 18÷54 =$$\frac{1}{3}$, $6÷18 = $$\frac{1}{3} and so on. The nth term rule is 54 ×$$\frac{1}{3}^n-1$.

$c.$ $0.01,0.1,1,10...$

$a = 0.01$ $r = 0.1$

The $nth$ term rule is $0.01 ×$$\frac{1}{10}^n-1$

• I hope this is right and sorry for the poor formatting. Apr 9, 2014 at 9:30