Live practice

Use the sequence definition or geometric-progression formula, keeping the first-term offset explicit.

The displayed substitutions cover every requested part and verify the result against the original data.

Proceed from the given expression and keep every transformation explicit.

The last line satisfies the required domain and establishes the result.

In a Cartesian product, the first coordinate comes from the first set.

\(G\times H=\{(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)\}\).

\(H\times G=\{(5,7),(5,8),(4,7),(4,8),(2,7),(2,8)\}\).

(a) For (mathrm{C_4H_8}) with one C=C bond, vary the position of the double bond and the carbon skeleton. The structures are (mathrm{CH_2=CHCH_2CH_3}) (but-1-ene), (mathrm{CH_3CH=CHCH_3}) (but-2-ene, which has cis and trans forms), and (mathrm{CH_2=C(CH_3)_2}) (2-methylpropene).

(b) For (mathrm{C_5H_8}) with one (mathrm{Cequiv C}) bond, the distinct structures are (mathrm{HCequiv CCH_2CH_2CH_3}) (pent-1-yne), (mathrm{CH_3Cequiv CCH_2CH_3}) (pent-2-yne), and (mathrm{HCequiv CCH(CH_3)CH_3}) (3-methylbut-1-yne).

Choose the longest chain containing the principal group and draw each carbon vertex once.

Track oxidation numbers on both sides. In every equation one element increases its oxidation number while another decreases, so oxidation and reduction occur together.

At total pressure 10⁵ Pa, p(I)=0.40P and p(I₂)=0.60P. Apply Kp=p(I)²/p(I₂).

Standard formation enthalpy uses each element in its most stable standard state as the reference.

Neutral main-group symbols show valence electrons. Add electrons for negative charge and remove them for positive charge.

The older law relates periodic properties to atomic mass. Moseley showed that nuclear charge, expressed by atomic number, is the fundamental ordering quantity.