Mdtryhht

$$a_n=frac117,frac10749,frac659343,frac48832401,frac3337116807,frac234569117649,dots$$

I don't know how to approach this question as it is not arithmetic or geometric. I know the denominator is geometric increasing by a factor of $7$ but I can't find what the numerator should be for the general formula for the terms. Anyone know what it is?

The general term seems to be
$$a_n=frac2(7^n)+(-3)^n7^n=2+left(-frac37right)^n$$
But the last term is given by
$$a_6=2-left(frac37right)^6$$
so this formula does not always work. A suitable formula could be
$$a_n=begincases2-left(frac37right)^n&nequiv0mod6\
2+left(-frac37right)^n &textotherwise
endcases$$

The general formula for an (infinite) sequence of (e. g. real) numbers from the finite number $n$ of its (first) members is in principle impossible, as the next (not listed) $(n+1)^mathrmth$ member may be an arbitrary number, and there is still a formula for expressing $a_1, dots, a_n, a_n+1,$ e. g. as a polynomial of order $n$:

$$a_k = sum_i=0^nb_ik^i,quad k = 1, dots,n+1$$

The process for finding coefficients $b_0, dots, b_n$ is straightforward enough.

In other words, if someone will find the formula for your "sequence", there is still the infinity number of other formulas, giving different sequences, but all of them producing your "sequence", i. e. $$frac117,frac10749,frac659343,frac48832401,frac3337116807,frac234569117649.$$

Note.

It means that all psychological tests of type

What is the next number of the sequence $1, 2, 3, 4, 5?$

are in principle meaningless ones, because you may tell "$1762$", and then show to surprised psychologist a formula supporting the correctness of your answer:

$$a_k = 439over30k^5-439over 2k^4+7463over 6k^3-6585over 2k^2+60158over 15k-1756$$

If he/she will not trust you, launch SageMath, which will produce accurate, non-rounded results, and write commands

sage: var("k")
sage: a(k) = (439/30)*k^5 - (439/2)*k^4+(7463/6)*k^3-(6585/2)*k^2+(60158/15)*k-1756
sage: a(1), a(2), a(3), a(4), a(5), a(6), a(7), a(8)

to obtain the result

(1, 2, 3, 4, 5, 1762, 10543, 36884)

(and to give the psychologist two more members for free).

The numerator of the $n$-th term seems to be
$$2cdot 7^n+(-1)^n 3^n. $$