how to prove that the sp3, sp2 and sp orbitals of CH_​3-CH=CH-CH₂-C=_C-CH₃? {C=_C is triple bond}

Dear Student,

For the given molecule,
$$\begin{gathered} C{H}_3-CH=CH-C{H}_2-C\equiv C-C{H}_3 \\ 1 2 3 4 5 6 7 \end{gathered}$$

There are three carbon atoms(1,4,7) which have a single bond to other carbon atoms. They are bonded to four entities. Thus, they are sp³ hybridized. There are four hybridized orbitals in sp³( s + 3p)
Total sp³ orbitals = 3 atoms x 4 orbitals = 12

There are two carbon atoms (2,3) which have one double bond between them and two single bonds to other atoms(C,H).
They follow sp² hybridization. There are three sp² orbitals.
Total sp² orbitals = 2 atoms x 3 orbitals = 6 orbitals

There are two carbon atoms(5,6) which have one triple bond between them and one single bond with other atoms(H).
They follow sp hybridizaton. There are two sp orbitals.
Total sp orbitals = 2 atoms x 2 orbitals = 4 orbitals

Ratio of number of orbitals is:
sp³ : sp² : sp = 12 : 6 : 4 = 6 : 3 : 2

Hope it helps.

Regards