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Number & Algebra

Chapter 1 · Sequences & Series

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Sequence
An ordered list of numbers u1,u2,u3,…u_1, u_2, u_3, \dots following a rule. Can be finite or infinite.
Series
The sum of a sequence: u1+u2+⋯+unu_1 + u_2 + \dots + u_n. Finite (sum to unu_n) or infinite (sum forever).
General term
unu_n is the nn-th term, expressed as a formula in nn. Two sequences differ iff their general terms do.
Arithmetic — definition
un+1−un=du_{n+1} - u_n = d
Common difference dd is constant across consecutive terms.
Arithmetic — $n$-th term
un=u1+(n−1)du_n = u_1 + (n-1)d
Arithmetic — sum
Sn=n2(u1+un)=n2(2u1+(n−1)d)S_n = \tfrac{n}{2}(u_1 + u_n) = \tfrac{n}{2}\bigl(2u_1 + (n-1)d\bigr)
Two terms gap
up−uq=(p−q)du_p - u_q = (p - q)d — the gap is exactly (p−q)(p-q) common differences.
Geometric — definition
un+1un=r\tfrac{u_{n+1}}{u_n} = r
Common ratio rr is constant. un≠0u_n \neq 0 required.
Geometric — $n$-th term
un=u1 r n−1u_n = u_1\,r^{\,n-1}
Geometric — finite sum
Sn=u1(1−rn)1−r=u1(rn−1)r−1  (r≠1)S_n = \frac{u_1(1 - r^n)}{1 - r} = \frac{u_1(r^n - 1)}{r - 1}\;(r\neq1)
Geometric — sum to infinity
S∞=u11−r    only if  ∣r∣<1S_\infty = \frac{u_1}{1-r}\;\;\text{only if}\;|r|<1
Convergent if ∣r∣<1|r|<1, divergent otherwise. Always check before applying.
Sigma notation
∑r=1nf(r)=f(1)+f(2)+⋯+f(n)\sum_{r=1}^{n} f(r) = f(1)+f(2)+\dots+f(n)
Sigma properties
∑(ar±br)=∑ar±∑br,    ∑c ar=c∑ar,    ∑r=1nc=cn\sum(a_r\pm b_r) = \sum a_r \pm \sum b_r,\;\;\sum c\,a_r = c\sum a_r,\;\;\sum_{r=1}^{n} c = cn
Standard sums
∑r=1nr=n(n+1)2,    ∑r=1nr2=n(n+1)(2n+1)6,    ∑r=1nr3=[n(n+1)2]2\sum_{r=1}^{n} r = \tfrac{n(n+1)}{2},\;\;\sum_{r=1}^{n} r^2 = \tfrac{n(n+1)(2n+1)}{6},\;\;\sum_{r=1}^{n} r^3 = \Bigl[\tfrac{n(n+1)}{2}\Bigr]^2
Term from $S_n$
un=Sn−Sn−1  (n≥2),    u1=S1u_n = S_n - S_{n-1}\;(n\ge 2),\;\;u_1 = S_1
Use when given SnS_n as a formula in nn.
Simple interest
FV=PV(1+rn100)FV = PV\Bigl(1 + \tfrac{rn}{100}\Bigr)
r%r\% per year, nn years; interest stays linear.
Compound interest
FV=PV(1+r100k)knFV = PV\Bigl(1 + \tfrac{r}{100k}\Bigr)^{kn}
kk = compounding periods per year. Per-period rate = r/(100k)r/(100k); periods = knkn.
Depreciation
FV=PV(1+r100)nFV = PV\Bigl(1 + \tfrac{r}{100}\Bigr)^n
Same shape; r<0r < 0 for depreciation.
Gauss's trick
Pair the 1st and last term, 2nd and 2nd-last, … each pair sums to u1+unu_1+u_n. With nn terms there are n/2n/2 pairs → derives Sn=n2(u1+un)S_n = \tfrac{n}{2}(u_1+u_n).
Find $d$ from two given terms
Use up−uq=(p−q)du_p - u_q = (p-q)d to solve for dd in ONE step.
Find $r$ from two given geometric terms
Divide them: up/uq=rp−qu_p/u_q = r^{p-q}. Take the appropriate root.
Common trap (sum to infinity)
∣r∣=1|r| = 1 does NOT converge (Grandi's series 1−1+1−1+…1-1+1-1+\dots is the classic warning). Always check ∣r∣<1|r| < 1 strictly.
Common trap (compound interest)
If interest compounds kk times per year, the EXPONENT is knkn (not nn); the per-period rate is r/(100k)r/(100k), not r/100r/100.