Q.The general formula for alkadiene is

A homologous series is a run of organic compounds that all share one characteristic functional group, and whose successive members differ from each other in molecular formula by exactly one $CH_2$ unit -- for example the alkanes (methane $CH_4$, ethane $C_2H_6$, propane $C_3H_8$...) or the alcohols (methanol $CH_3OH$, ethanol $C_2H_5OH$, propanol $C_3H_7OH$...).

Because every member of a series shares the same functional group and the same $CH_2$-unit spacing, the entire series can be captured by ONE general formula: alkanes are $C_nH_{2n+2}$, alkenes are $C_nH_{2n}$, alkynes are $C_nH_{2n-2}$ -- and since an alkadiene (a chain with two C=C double bonds) has the same degree of unsaturation as an alkyne (one C≡C triple bond, or two degrees of unsaturation total), alkadienes share the alkyne formula, $C_nH_{2n-2}$. This same general-formula logic extends class by class: an aliphatic monohydric alcohol is $C_nH_{2n+1}OH$ (one -OH on a saturated chain); aliphatic ketones are $C_nH_{2n}O$ (one C=O on a saturated chain, needing at least 3 carbons since the carbonyl carbon needs two carbon substituents); aliphatic amines are $C_nH_{2n+3}N$ (one -NH2, or a substituted nitrogen, on a saturated chain); nitroalkanes are $C_nH_{2n+1}NO_2$ for the first six members: $CH_3NO_2$ (nitromethane), $C_2H_5NO_2$ (nitroethane), $C_3H_7NO_2$ (nitropropane), $C_4H_9NO_2$ (nitrobutane), $C_5H_{11}NO_2$ (nitropentane), $C_6H_{13}NO_2$ (nitrohexane).

The carboxylic acid series ($C_nH_{2n}O_2$, or $C_nH_{2n+1}COOH$) is worth walking through by name and structure for its first four members: formic acid, $HCOOH$ ($CH_2O_2$); acetic acid, $CH_3COOH$ ($C_2H_4O_2$); propanoic acid, $CH_3CH_2COOH$ ($C_3H_6O_2$); and butanoic acid, $CH_3CH_2CH_2COOH$ ($C_4H_8O_2$) -- each member simply one $CH_2$ longer than the last, all sharing the -COOH functional group and the -oic acid suffix.

Because every member of a homologous series shares one functional group, they also share almost identical CHEMICAL behaviour (the same characteristic reactions, happening at the same functional group), even while their PHYSICAL properties (melting point, boiling point, density, solubility) grade smoothly and predictably as the chain lengthens -- and members of a series can typically all be prepared by the same general synthetic methods, which is part of why organising compounds into homologous series is such a powerful simplification.