Find the nth term of $2, -6, -4$

Question

I have been solving arithmetic and geometric progression and stumbled onto this :

Find the nth term of $2, -6, -4$

The answer is $5 - 3n$ from $A + Bn = 5 - 3n$ where $A = 5$ and $B = -3$

I can't seem to get or solve how did it reached to that answer. There were no pattern to the sequence.

I tried the following :

• Substitute the values of $n$ to the final answer. $5 - 3(1) = 2$ which is correct, as $2$ is the 1st term. But when you substitute $n = 2; 5 - 3(2) = -1$ which is not correct since the $2^{nd}$ term is $-6$. Then substitute $n = 3; 5 - 3(3) = -4$ which is correct.

• The book provides a solution via calculator technique. By listing the values on the table, it is found that $A = 5$. But when I do the same exact method he did, the A is equal to $3.3333$ and not $5$. Looking for $B$ via calculator seems to be correct which is $-3$.

• I can't do manual solving as n is not given and the difference (If its an arithmetic) or ratio (if its geometric) is not given and can't be analyze.

Any help would be appreciated. Thank you

Answer 1

"I have been solving arithmetic and geometric progression and stumbled onto this"

"Find the nth term of 2,−6,−4"

Okay, presumably this is either arithmetic or geometric.

If arithmetic then $a_n = a_0 + d*n$. and $a_1 = 2$. $a_2 = -6 = a_1 + d = 2 + d$ so $d=-8$. If we verify this with $a_3 = -4 = a_2 + d = -6 -8 = -14$. This doesn't fit and it isn't an arithmetic progression at all.

Actually, That $a_1 > a_2$ and $a_3< a_4$ should have been a tip off. All arithmetic progressions are increasing, decreasing, or constant. This being neither of the three can't be arithmetic.

The same is tre of geometric sequence. A geometric sequence is $a_n = a_1*b^n$ and if $b>1$ is increasing and if $b < 1$ is decreasing. If this were geometric we would have $\frac {a_1}{a_2} = \frac {a_2}{a_3}$ which clearly doesn't happen. $\frac {2}{-6} = -\frac 13$ but $\frac {-6}{-4} = \frac 32$.

"The answer is 5−3n from A+Bn=5−3n where A=5 and B=−3"

Well, that's just nutty.

$2 = 5 - 3*1$ so that's okay for the first term but $-6 \ne 5 - 3*2$. However $-4 = 5 - 3*3 $.

So, clearly the text has a typo and they meant what is $n$th term of $2,-1,-4$. And that would be the arithmetic sequence $a_n = 5-3n$.

Answer 2

It's either there is a typo in the question or the answer is wrong.

Notice that the squence is not even monotone, hence it can't be an airthmetic sequence.

Answer 3

There is no sensible answer to the question as there are too few terms given to establish a pattern. It is neither arithmetic nor geometric. Many on this site will tell you there is no sensible answer to any of these problems, but I believe if there is a pattern that is clearly established you should follow it.

Answer 4

The correct sequence is $2,-1,-4$.

Taking pairwise differences, you find $-3,-3$ which is the common difference (and indeed constant). Then $a_0+(-3)\cdot1=2$

leads you to

$$a_n=5-3n.$$