The carbonate ion, (CO_{3}^{2-}), has three resonance structures with a central carbon and three oxygens in trigonal planar geometry; all C–O bonds are equivalent with bond order 4/3.
CO₃²⁻ has a central carbon bonded to three oxygens; in one Lewis form one O is double-bonded and the other two are single-bonded with a −1 charge each (three equivalent resonance forms). ASCII (one resonance form): O || O^- – C – O^- Note: the other two forms rotate which O is the double-bonded one.
HCO3− has a central C with a C=O, a C–OH, and a C–O⁻; the negative charge is delocalized between the two O atoms via resonance (HO–C(=O)–O⁻ and HO–C(−O⁻)=O).
CO_3^{2-} (carbonate) has three resonance structures: one C=O and two C–O⁻ bonds; the negative charge is delocalized over all oxygens, giving each C–O bond a bond order of 4/3 (≈1.33).
Carbonate ion, CO₃²⁻, has a central C atom bonded to three O atoms with three equivalent resonance structures; each structure has one C=O and two C–O⁻ bonds, giving an average bond order of 1.33.
HCO₃⁻ has a central carbon with three oxygens: one C=O, one C–OH, and one C–O⁻. The negative charge is delocalized over the two single-bonded oxygens via resonance, giving three equivalent structures.
Bicarbonate (HCO₃⁻) Lewis structure: carbon in the center with a C=O double bond to one O, a C–OH single bond to another O, and a C–O⁻ single bond to the third O; the negative charge is delocalized over the two singly bonded oxygens (resonance).
CO3^2- (carbonate ion) has a central carbon atom bonded to three oxygens. In resonance, one C=O double bond and two C-O^- single bonds share the -2 charge delocalized over the three oxygens (three equivalent resonance forms).
Continuously compounded interest is interest calculated an infinite number of times per period. The balance after time t on principal P at rate r is $$A = P e^{rt}$$.
Continuous compounding uses the exponential formula $$A = P e^{rt}$$. Here P is the principal, r is the annual rate, t is time in years, and A is the final amount.
Continuous compounding means interest is compounded continuously; the amount after t years is $$A = P e^{rt}$$.
Continuous interest, or continuous compounding, is interest that accrues continuously; the amount is $$A = P e^{rt}$$.
Continuous compounding means interest is calculated an infinite number of times per period, so A = P e^{rt}.
The continuous-compounding formula is A = P e^{rt}. Key Concepts Concept: Formula form: A = P e^{rt} Concept: Variables: P, r, t Concept: Interest I = A − P Steps Action: Compute A using A = P e^{rt} Action: Solve for r: r = frac{1}{t} lnleft(frac{A}{P}right) Action: Compute interest: I = A − P Formula […]
Interest is compounded continuously, so $$A = P e^{rt}$$, where P is principal, r the annual rate, and t the time in years.
A = P e^{rt} is the continuous-compounding formula. Key Concepts Concept: Continuous compounding Concept: Accumulation factor e^{rt} Concept: Principal vs final amount Steps Action: Identify P, r, t values Action: Compute exponent rt Action: Evaluate A = P e^{rt} Formula $$A = P e^{rt}$$ $$I = A – P = Pleft(e^{rt} – 1right)$$
A(t) = P e^{rt} for continuous compounding. Key Concepts Concept: Continuous compounding grows exponentially Concept: Force of interest δ Concept: Link to effective rate i Steps Action: Model with dA/dt = rA Action: Solve by separation of variables Action: Apply A(0)=P to get A(t) Formula $$A(t) = P e^{rt}$$ $$A(t) = P e^{delta t} quad […]
A = P e^{rt}, where P is principal, r is rate, t is time. Key Concepts Concept: Continuous compounding growth model Concept: Principal P Concept: Final amount A = P e^{rt} Steps Action: Identify P, r, t Action: Compute e^{rt} (exponential factor) Action: Multiply by P to get A Formula $$A = P e^{rt}$$ Example […]
Average the absolute deviations from a chosen center (mean or median). Key Concepts Center choice: mean or median Deviation: absolute difference |x_i − center| MAD variants: mean-based or median-based MAD Steps Action: Choose center (mean or median) Action: Compute d_i = |x_i − center| Action: Compute MAD by chosen method Formula MAD from mean: $$MAD_{text{mean}} […]