Let us define a sequence (an)n≥1(a_n)_{n \geq 1}(an)n≥1 of real numbers to be a Fibonacci-like sequence if an=an−1+an−2,n≥3.a_n = a_{n-1} + a_{n-2}, \quad n \geq 3.an=an−1+an−2,n≥3.What is the dimension of the R$$\mathbb{R}$$R -vector space of Fibonacci-like sequences?

If an=an−1+an−2, n≥3a_n = a_{n-1} + a_{n-2}, \\; n \\geq 3an=an−1+an−2,n≥3 ,the terms can be written as:a3=a2+a1a4=a3+a2a5=a4+a3⋮an=an−1+an−2.a_3 = a_2 + a_1 \\\\[10pt]a_4 = a_3 + a_2 \\\\[10pt]a_5 = a_4 + a_3 \\\\\\vdots\\\\a_n = a_{n-1} + a_{n-2} .a3=a2+a1a4=a3+a2a5=a4+a3⋮an=an−1+an−2.Substituting values, the equations become:a3=a2+a1,a4=a2+a1+a2a5=a2+a1+a2+a2,⋮a_3 = a_2 + a_1,\\\\[10pt]a_4 = a_2 + a_1 + a_2\\\\[10pt]a_5 = a_2 + a_1 + a_2 + a_2,\\\\\\vdotsa3=a2+a1,a4=a2+a1+a2a5=a2+a1+a2+a2,⋮and so on.As all terms can be generated by a1a_1a1 and a2a_2a2, the dimension of the vector space will be 2. ⟹ Option (B) is correct.\\implies $$\\text{Option (B) is correct.}$$⟹Option (B) is correct.