Find S12 for the series 1 + 1.5 + 2.25 + 3.375 +…

This is a geometric series.....the sum of the first n terms is given by

S_n = a(1 - r^n) / ( 1 - r)

Where a is the first term, r is the common ratio,  and n is the number of terms we wish to sum

So, here  

n = 12  a = 1 and r = 1.5

So we have

S₁₂ = 1(1 - 1.5^12) /(1 - 1.5)   = 257.49267578125