Valid term from quadratic sequence?
JavaScript (ES7), 70 bytes
Returns a Boolean value.
(a,b,c,t)=>(t-=c,(a*=2)?(x=(b*b+2*a*t)**.5-b)%a&&(x+b+b)%a:b?t%b:t)==0
Try it online!
How?
For the sake of clarity, we define \$d = T_n-c\$. (The same variable \$t\$ is re-used to store this result in the JS code.)
Case \$a\neq0\$
The equation really is quadratic:
$$T_n=an^2+bn+c\\ an^2+bn-d=0$$
With \$a'=2a\$, the discriminant is:
$$\Delta=b^2+2a'd$$
and the roots are:
$$n_0=\frac{-b-\sqrt{\Delta}}{a'}\\ n_1=\frac{-b+\sqrt{\Delta}}{a'}$$
The equation admits an integer root if \$\sqrt{\Delta}\$ is an integer and either:
$$\begin{align}&-b-\sqrt{\Delta}\equiv 0\pmod{a'}\\ \text{ or }&-b+\sqrt{\Delta}\equiv 0\pmod{a'}\end{align}$$
Case \$a=0, b\neq0\$
The equation is linear:
$$T_n=bn+c\\ bn=d\\ n=\frac{d}{b}$$
It admits an integer root if \$d\equiv0\pmod b\$.
Case \$a=0, b=0\$
The equation is not depending on \$n\$ anymore:
$$T_n=c\\ d=0$$
Jelly, 11 10 bytes
_/Ær1Ẹ?%1Ạ
A monadic Link which accepts a list of lists* [[c, b, a], [T_n]] and yields 0 if T_n is a valid solution or 1 if not.
* admittedly taking a little liberty with "You may take input of these four numbers in any way".
Try it online! Or see a test-suite.
How?
_/Ær1Ẹ?%1Ạ - Link: list of lists of integers, [[c, b, a], [T_n]]
/ - reduce by:
_ - subtraction [c-T_n, b, a]
? - if...
Ẹ - ...condition: any?
Ær - ...then: roots of polynomial i.e. roots of a²x+bx+(c-T_n)=0
1 - ...else: literal 1
%1 - modulo 1 (vectorises) i.e. for each: keep any fractional part
- note: (a+bi)%1 yields nan which is truthy
Ạ - all? i.e. all had fractional parts?
- note: all([]) yields 1
If we could yield non-distinct results then _/Ær1Ẹ?ḞƑƇ would also work for 10 (it yields 1 when all values are solutions, otherwise a list of the distinct solutions and hence always an empty list when no solutions - this would also meet the standard Truthy vs Falsey definition)