Sequence & Series — NDA Mathematics

Formulas (8)

• Sequence, series, and the nth term · The two bridges between term and sum
  $$a_n = S_n - S_{n-1}\quad (n \ge 2), \qquad a_1 = S_1$$
• nth term and sum of an AP · nth term and sum
  $$a_n = a + (n-1)d, \qquad S_n = \frac{n}{2}\,[\,2a + (n-1)d\,] = \frac{n}{2}(a + l)$$
• Recovering the term from a sum-formula · Term from sum
  $$a_n = S_n - S_{n-1}\quad (n \ge 2)$$
• The arithmetic mean and symmetric terms · Arithmetic mean of a and b
  $$\text{AM} = \frac{a + b}{2}$$
• The three-term condition and what preserves an AP · Three terms in AP
  $$a,\ b,\ c \text{ in AP} \iff 2b = a + c$$
• Ratio of sums and ratio of terms · Ratio of nth terms from ratio of sums
  $$\frac{a_n}{a_n'} = \frac{f(2n-1)}{g(2n-1)}\quad\text{when}\quad \frac{S_n}{S_n'} = \frac{f(n)}{g(n)}$$
• The clever AP identities · Clever AP identities
  $$S_m = n,\ S_n = m \Rightarrow S_{m+n} = -(m+n) \qquad p\,a_p = q\,a_q \Rightarrow a_{p+q} = 0 \qquad S_p = S_q \Rightarrow S_{p+q} = 0$$
• Common terms of two APs · Common-terms AP
  $$d_{\text{common}} = \operatorname{lcm}(d_1, d_2)$$

Watch out for (4)

• The nth term uses $(n-1)d$, not $nd$
  → nth term and sum of an AP
• The sum has $(n-1)d$ inside the bracket
  → nth term and sum of an AP
• Inserting $k$ means makes $k+1$ gaps, not $k$
  → The arithmetic mean and symmetric terms
• Squaring breaks the AP
  → The three-term condition and what preserves an AP

Formulas (5)

• nth term, geometric mean, and the three-term condition · nth term and three-term condition
  $$a_n = a\,r^{\,n-1}, \qquad b^2 = ac \ \text{(for } a,b,c \text{ in GP)}$$
• Sum of a finite GP · Sum of n terms
  $$S_n = \frac{a\,(r^n - 1)}{r - 1}\quad (r \ne 1)$$
• Sum of an infinite GP · Sum to infinity
  $$S_\infty = \frac{a}{1 - r}\quad (|r| < 1)$$
• Periodic continued fractions & nested radicals · Self-referential equations
  $$x = a + \tfrac{1}{x}\ \Rightarrow\ x^2 - ax - 1 = 0, \qquad x = \sqrt{a + x}\ \Rightarrow\ x^2 - x - a = 0$$
• Product of terms and the middle-term trick · GP product symmetry
  $$a_k \cdot a_{n+1-k} = a_1 \cdot a_n \qquad \prod_{i=1}^{2m-1} a_i = M^{2m-1}$$

Watch out for (5)

• The GP nth term is $a\,r^{n-1}$, not $a\,r^{n}$
  → nth term, geometric mean, and the three-term condition
• Repeating-digit sums hide a GP
  → Sum of a finite GP
• The convergence condition is not optional
  → Sum of an infinite GP
• It is $\dfrac{a}{1-r}$, watch the sign in the denominator
  → Sum of an infinite GP
• It is NOT an infinite GP
  → Periodic continued fractions & nested radicals

Formulas (3)

• Harmonic progression — flip to the reciprocal AP · HP nth term and three-term condition
  $$a_n = \frac{1}{a + (n-1)d}, \qquad b = \frac{2ac}{a+c}\ \text{(for } a,b,c \text{ in HP)}$$
• AM, GM, HM and the inequality that orders them · The three means and their relation
  $$\text{AM} = \frac{a+b}{2},\quad \text{GM} = \sqrt{ab},\quad \text{HM} = \frac{2ab}{a+b}, \qquad \text{GM}^2 = \text{AM}\cdot\text{HM}$$
• Harmonic mean of several numbers · Harmonic mean of n numbers
  $$\text{HM} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$$

Watch out for (4)

• You cannot average HP terms directly
  → Harmonic progression — flip to the reciprocal AP
• AM ≥ GM ≥ HM only for positives
  → AM, GM, HM and the inequality that orders them
• Don't swap the three mean formulas
  → AM, GM, HM and the inequality that orders them
• The identity is $\text{GM}^2 = \text{AM}\cdot\text{HM}$
  → AM, GM, HM and the inequality that orders them

Formulas (4)

• The three three-term conditions · The unifying ratio
  $$\frac{a-b}{b-c} = \begin{cases} 1 & \text{AP} \\[2pt] a/b & \text{GP} \\[2pt] a/c & \text{HP} \end{cases}$$
• The log bridge: a GP becomes an AP · The bridge
  $$x, y, z \text{ in GP} \iff \log x, \log y, \log z \text{ in AP}$$
• The reciprocal bridge: an HP becomes an AP · Reciprocal flip
  $$\frac{1}{u}, \frac{1}{v}, \frac{1}{w} \text{ in HP} \iff u, v, w \text{ in AP}$$
• Roots, coefficients, and progression conditions · Vieta's relations (monic-friendly)
  $$\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}$$

Formulas (3)

• Sums of powers of natural numbers · The three power sums
  $$\sum k = \frac{n(n+1)}{2},\quad \sum k^2 = \frac{n(n+1)(2n+1)}{6},\quad \sum k^3 = \left[\frac{n(n+1)}{2}\right]^2$$
• Factorial sums — telescoping and remainders · Factorial telescoping
  $$n\cdot n! = (n+1)! - n! \;\Rightarrow\; \sum_{k=1}^{n} k\cdot k! = (n+1)! - 1$$
• Telescoping sums, repunits, and divisibility patterns · Telescoping standard sum
  $$\sum_{k=1}^{n} \frac{1}{k(k+1)} = 1 - \frac{1}{n+1} = \frac{n}{n+1}$$

Watch out for (1)

• Don't confuse the three power-sum formulas
  → Sums of powers of natural numbers