Step-by-step explanation:

[tex]\rm d = a_3 - a_2[/tex]

[tex]\rm = 299413620[/tex]

[tex]\rm a_n = a+(n-1)d[/tex]

[tex]\rm a_5 = 10+4×299413620[/tex]

[tex]\rm a_5 = 1197654490[/tex]

Verified answer

[tex]\rm a_{n+1}=a_{n}(a_{n}^{2}-3a_{n}+3)[/tex]

[tex]\;[/tex]

[tex]\rm a_{n+1}-1=a_{n}^{3}-3a_{n}^{2}+3a_{n}-1[/tex]

[tex]\;[/tex]

We know,

[tex]\boxed{\rm{(A+B)^{3}=A^{3}+3A^{2}{B}+3AB^{2}+B^{3}}}[/tex]

[tex]\rm a_{n+1}-1=(a_{n}-1)^{3}[/tex]

[tex]\;[/tex]

[tex]\rm \log_{3}(a_{n+1}-1)=\log_{3}(a_{n}-1)^{3}[/tex]

[tex]\;[/tex]

We know,

[tex]\boxed{\rm\log_{a}m^{n}=n\log_{a}m}[/tex]

[tex]\rm\log_{3}(a_{n+1}-1)=3\log_{3}(a_{n}-1)[/tex]

[tex]\;[/tex]

[tex]\rm\log_{3}(a_{n}-1)[/tex] is a geometric sequence.[tex]\;[/tex]

Properties of the sequence

• Common ratio: 3
• The first term: 2

[tex]\rm\therefore\log_{3}(a_{n}-1)=2\cdot3^{n-1}[/tex]

[tex]\;[/tex]

For [tex]\rm n=5[/tex],

[tex]\rm\log_{3}(a_{5}-1)=2\cdot3^{4}[/tex]

[tex]\;[/tex]

[tex]\rm\log_{3}(a_{5}-1)=162[/tex]

[tex]\;[/tex]

[tex]\rm a_{5}-1=3^{162}[/tex]

[tex]\;[/tex]

[tex]\rm\therefore a_{5}=\boxed{3^{162}+1}[/tex]