IB Maths AA · Topic 1 · 32 cards

Number and algebra flashcards

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What is the formula for the \(n\)th term of an arithmetic sequence?
\(u_n=u_1+(n-1)d\), where \(u_1\) is the first term and \(d\) is the common difference (\(d=u_{n+1}-u_n\)).
What are the two formulas for the sum of the first \(n\) terms of an arithmetic series?
\(S_n=\frac{n}{2}\big(2u_1+(n-1)d\big)\) or \(S_n=\frac{n}{2}(u_1+u_n)\). Use the second when the last term is known.
What is the formula for the \(n\)th term of a geometric sequence?
\(u_n=u_1r^{\,n-1}\), where \(r\) is the common ratio (\(r=u_{n+1}\div u_n\)).
What is the formula for the sum of the first \(n\) terms of a geometric series?
\(S_n=\frac{u_1(r^n-1)}{r-1}=\frac{u_1(1-r^n)}{1-r}\), for \(r\neq 1\).
When does a geometric series have a sum to infinity, and what is it?
It converges only when \(|r|<1\). Then \(S_\infty=\frac{u_1}{1-r}\). If \(|r|\ge 1\) the terms do not shrink to zero, so there is no finite sum.
How can you tell whether a sequence is arithmetic or geometric?
Arithmetic: consecutive terms differ by the same amount (constant difference). Geometric: consecutive terms have the same ratio. Test two or three consecutive pairs.
Calculation: find \(S_\infty\) for the series \(12+4+\frac43+\dots\)
\(r=\frac13\), so \(S_\infty=\frac{12}{1-\frac13}=\frac{12}{\frac23}=18\).
Calculation: an arithmetic series has \(u_1=7\) and \(d=4\). Find \(S_{20}\).
\(S_{20}=\frac{20}{2}\big(2(7)+19(4)\big)=10\times90=900\).
What does \(\sum_{r=1}^{n} u_r\) mean?
The sum of the terms \(u_r\) as \(r\) runs from 1 to \(n\): \(u_1+u_2+\dots+u_n\). The lower value is where you start; the upper value is where you stop.
State the laws of exponents for \(a^m\times a^n\), \(a^m\div a^n\) and \((a^m)^n\).
\(a^m a^n=a^{m+n}\); \(a^m\div a^n=a^{m-n}\); \((a^m)^n=a^{mn}\). Also \(a^0=1\), \(a^{-n}=\frac1{a^n}\) and \(a^{1/n}=\sqrt[n]{a}\).
How do you solve \(8^{x}=32\) without logarithms?
Write both sides with the same base: \(2^{3x}=2^{5}\), so \(3x=5\) and \(x=\frac53\).
What is the definition of a logarithm?
\(\log_a b=x\) means \(a^x=b\) (with \(a>0\), \(a\neq1\), \(b>0\)). Logarithms undo exponentials: \(a^{\log_a x}=x\). \(\ln x\) is \(\log_e x\).
State the three laws of logarithms.
\(\log_a(xy)=\log_a x+\log_a y\); \(\log_a\!\left(\frac xy\right)=\log_a x-\log_a y\); \(\log_a(x^k)=k\log_a x\).
What is the change of base formula for logarithms?
\(\log_a b=\frac{\log_c b}{\log_c a}\) for any valid base \(c\). It lets you evaluate any logarithm using \(\ln\) or \(\log_{10}\) on a calculator.
Calculation: solve \(2^x=50\), giving the answer to 3 s.f.
\(x=\log_2 50=\frac{\ln 50}{\ln 2}\approx 5.64\).
What is the compound interest formula, and what do the symbols mean?
\(FV=PV\left(1+\frac{r}{100k}\right)^{kn}\): \(PV\) is the starting amount, \(r\) the annual rate in %, \(k\) the number of compounding periods per year, \(n\) the number of years.
Calculation: €2000 is invested at 3% per year compounded annually. Find its value after 5 years.
\(2000\times1.03^{5}=2318.55\), so the value is about €2318.55.
How is a number written in standard form?
\(a\times10^{k}\) with \(1\le a<10\) and \(k\) an integer. For example \(0.00042=4.2\times10^{-4}\) and \(53\,000=5.3\times10^{4}\).
Calculation: a car worth €18 000 loses 15% of its value each year. Find its value after 4 years.
Each year the value is multiplied by \(0.85\). \(V=18\,000\times0.85^{4}=9396.11\), so about €9396.
What does the binomial theorem say for \((a+b)^n\) with \(n\) a positive integer?
\((a+b)^n=\sum_{r=0}^{n}\binom nr a^{\,n-r}b^{\,r}\). The general term is \(\binom nr a^{n-r}b^{r}\) and \(\binom nr=\frac{n!}{r!(n-r)!}\).
Calculation: find the coefficient of \(x^3\) in the expansion of \((2+x)^5\).
General term \(\binom5r 2^{5-r}x^r\). With \(r=3\): \(\binom53\times2^2=10\times4=40\).
(HL) What are the steps of a proof by induction?
1. Base case: show the statement holds for the first value (usually \(n=1\)). 2. Assume it holds for \(n=k\). 3. Show it then holds for \(n=k+1\). 4. Conclude it holds for all \(n\) by induction.
(HL) How do you find the modulus and conjugate of \(z=a+bi\)?
\(|z|=\sqrt{a^2+b^2}\) and \(z^*=a-bi\). Useful fact: \(zz^*=a^2+b^2=|z|^2\), which is real.
(HL) What are the polar and Euler forms of a complex number?
\(z=r(\cos\theta+i\sin\theta)=r\,\mathrm{cis}\,\theta=re^{i\theta}\), where \(r=|z|\) and \(\theta=\arg z\) is the angle from the positive real axis.
(HL) How do you multiply and divide complex numbers in modulus-argument form?
Multiply the moduli and add the arguments: \(z_1z_2=r_1r_2\,\mathrm{cis}(\theta_1+\theta_2)\). Divide the moduli and subtract the arguments: \(\frac{z_1}{z_2}=\frac{r_1}{r_2}\mathrm{cis}(\theta_1-\theta_2)\).
(HL) State de Moivre's theorem.
\([r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)\) for integer \(n\). In Euler form: \((re^{i\theta})^n=r^ne^{in\theta}\).
(HL) Calculation: evaluate \((1+i)^8\).
\(|1+i|=\sqrt2\) and \(\arg=\frac\pi4\). So \((1+i)^8=(\sqrt2)^8\,\mathrm{cis}(2\pi)=16\).
(HL) How do you find the \(n\)th roots of a complex number \(z=re^{i\theta}\)?
The roots are \(r^{1/n}e^{i(\theta+2\pi k)/n}\) for \(k=0,1,\dots,n-1\). They lie on a circle of radius \(r^{1/n}\), equally spaced by \(\frac{2\pi}{n}\).
(HL) What is the conjugate root theorem?
If a polynomial has real coefficients and \(a+bi\) is a root, then \(a-bi\) is also a root. So non-real roots come in conjugate pairs.
(HL) How do you split \(\frac{1}{(x-1)(x+2)}\) into partial fractions?
Write \(\frac{A}{x-1}+\frac{B}{x+2}\) and equate \(1=A(x+2)+B(x-1)\). Then \(x=1\) gives \(A=\frac13\) and \(x=-2\) gives \(B=-\frac13\).
(HL) How does the binomial expansion extend to \((1+x)^n\) when \(n\) is negative or a fraction?
\((1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\dots\), an infinite series that is valid only for \(|x|<1\).
(HL) How can a system of three linear equations in three unknowns behave?
It can have a unique solution (planes meet at one point), infinitely many solutions (a line or a plane in common), or no solution (inconsistent). Solve using a GDC or row reduction.

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