what is the formula for the sequence 1,9,26...
You only have three numbers here and they do not fit either an arithmetic sequence or a geometric sequence.
Excepting the above, three numbers do not a sequence make (usually). In order to establish the sequence, you need to have enough terms in your sequence for analysis to yield a pattern.
One thing about apparently unrelated numbers though, they are still subject to a mathematical analysis, in the sense that ...
Two numbers can be fitted to a straight line – linear equation
Three numbers can be fitted to a quadratic equation
Four numbers can be fitted to a cubic equation
And so on ...
What we have are three numbers: 1, 9, 26.
Let us fit these numbers to a quadratic equation, y = ax^2 + bx + c.
Taking the points (1, 1), (2, 9), (3, 26) to lie on the quadratic equation, then we can write ...
1 = a.1^2 + b.1 + c
9 = a.2^2 + b.2 + c
26 = a.3^2 + b.3 + c
These eqns become
1 = a + b + c ---------------------- (1)
9 = 4a + 2b + c ------------------- (2)
26 = 9a + 3b + c ----------------- (3)
Subtracting (1) from (2), and (2) from (3) gives us,
8 = 3a + b ------------------------- (4)
17 = 5a + b ----------------------- (5)
Subtracting (4) from (5),
9 = 2a
a = 4.5
Substituting for a = 4.5 into (4),
8 = 13.5 + b
b = -5.5
Substituting for a = 4.5 and b = -5.5 into (1),
1 = 4.5 – 5.5 + c
c = 2
The quadratic equation then is,
y = 4.5x^2 – 5.5x + 2
Converting this into the generator for our sequence,
t_n = 4.5n^2 – 5.5n + 2
where t_n is the nth term in the sequence.