Sequences and series

Explanation

• For a positive integer $n$, $(a+bx)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}(bx)^r$, where $\binom{n}{r}=n!/[r!(n-r)!]$, $n!=n(n-1)\cdots1$ and $0!=1$; Pascal's relation is $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-1}{r}$ for $1\le r\le n-1$. The same coefficients appear in binomial probabilities: $P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}$ for $X\sim B(n,p)$.
• For rational $n$, write the expression as $a^n(1+u)^n$ and use $1+nu+n(n-1)u^2/2!+\cdots$, valid for $|u|<1$.
• For an approximation, choose a nearby convenient value and retain the requested number of terms.
• A common error is to omit powers of the coefficient $b$ from terms involving $(bx)^r$.
• Use $(1+u)^n=1+nu+n(n-1)u^2/2+\cdots$ with $n=-1/2$ and $u=-2x$.

Explanation

• An explicit rule gives $x_n$ directly from $n$, whereas a recurrence relation defines each term from one or more preceding terms and needs an initial value.
• A sequence is increasing when $x_{n+1}>x_n$ and decreasing when $x_{n+1}<x_n$ throughout the stated range of $n$.
• A sequence is periodic if its terms repeat after a fixed positive number of steps; the least such number is its period.
• When using a recurrence, retain sufficient accuracy between steps and check any invariant interval.
• A common error is to apply $f$ repeatedly to the initial term instead of to the latest term.

Explanation

• The notation $\sum_{r=a}^{b}f(r)$ means add the values of $f(r)$ for each integer $r$ from $a$ to $b$ inclusive.
• To write a series in sigma notation, identify a formula for its general term together with the correct first and last index values.
• Use linearity to split sums: for example, $\sum_{r=1}^{n}(ar+b)=a\sum_{r=1}^{n}r+b\sum_{r=1}^{n}1$, with $\sum_{r=1}^{n}1=n$.
• Check the endpoint terms after changing an index.
• A common error is to treat the upper limit as the number of terms when the lower limit is not $1$.

Explanation

• An arithmetic sequence has constant difference $d$ and nth term $u_n=a+(n-1)d$, where $a$ is the first term.
• The sum of its first $n$ terms is $S_n=n[2a+(n-1)d]/2$, equivalently $S_n=n(a+l)/2$ when the last term $l$ is known.
• Pairing the first and last terms, then the second and penultimate terms, gives $n$ equal pairs of total $a+l$ across two copies of the series, proving $S_n=n(a+l)/2$.
• The term number $n$ must be a positive integer.
• A common error is to use $a+nd$ for the nth term, shifting every term by one difference.

Explanation

• A geometric sequence has constant ratio $r$ and nth term $u_n=ar^{n-1}$, where $a$ is the first term.
• For $r\ne1$, subtracting $rS_n$ from $S_n$ proves $S_n=a(1-r^n)/(1-r)$; an equivalent form may be more convenient when $r>1$.
• A geometric series has a finite sum to infinity only when $|r|<1$, in which case $S_\infty=a/(1-r)$; for a finite-sum threshold, isolate $r^n$ and use logarithms.
• A negative ratio makes term signs alternate, but convergence still depends on its modulus.
• A common error is to use $r<1$ instead of $|r|<1$, which would wrongly accept ratios below $-1$.

Explanation

• A constant additive change suggests an arithmetic model, while a constant multiplier or percentage change suggests a geometric model.
• State what the term number represents and whether the initial value is $u_1$ or $u_0$ before forming a term or sum.
• Repeated deposits, withdrawals or other fixed adjustments can be represented by a recurrence and often rewritten using a finite geometric sum.
• Interpret results within the context and identify assumptions such as fixed rates or indefinite continuation.
• A common error is to use an infinite sum when the process has not converged or has a finite stopping point.