Revision sheet

Linear or exponential?

Equal differences: linear. Equal factors: exponential.

Watch out: exponential growth wins in the end.

Arithmetic sequences

• Add $d$ each time: $a_n = a_1 + (n - 1)d$.
• $d = \dfrac{a_q - a_p}{q - p}$.

Watch out: $n - 1$ steps from $a_1$ to $a_n$.

Geometric sequences

• Multiply by $r$ each time: $a_n = a_1 r^{\,n-1}$.
• $r$ = a term divided by the one before.

Watch out: the exponent is $n - 1$.

Rational exponents

• $b^{\frac{1}{n}} = \sqrt[n]{b}$; $b^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m$.
• $b^{-r} = \dfrac{1}{b^{r}}$: the value stays positive.
• A factor $b$ per unit means a factor $b^{\frac12}$ per half-unit.

Exponential change: the same factor at every step

• Start $a$, factor $b$ per step: after $n$ steps, $a \cdot b^n$.
• Gain $p\%$: factor $1 + \frac{p}{100}$. Lose $p\%$: factor $1 - \frac{p}{100}$.

Watch out: adding the same amount each step is linear, not exponential.

Reading exponential functions

$a$: value at $x = 0$; $b$: factor per step; $b^{-1} = 1/b$.

Watch out: negative exponent, positive value.

Writing an exponential function

• $f(x) = a \cdot b^x$: $a = f(0)$, $b = \dfrac{f(1)}{f(0)}$.
• Multiplied by $b$ every $n$ units: $f(x) = a \cdot b^{\frac{x}{n}}$.

Watch out: the starting value is $f(0)$, and it goes in front, not in the base.

Percent change and interest

Increase $r$: × $(1 + r)$; decrease: × $(1 - r)$; compounding: rate and exponent per period.

Watch out: 1.1, not 0.1, for a 10% increase.