Sequence and Series: AFCAT Numerical Ability

Formulas (6)

• The nth term of an AP · nth term of an AP
  $$a_n = a + (n - 1)d$$
• Which term is it, and how many terms are there · Number of terms
  $$n = \dfrac{l - a}{d} + 1$$
• An AP from two of its terms · Difference from two terms
  $$d = \dfrac{a_q - a_p}{q - p}$$
• Sum of the first n terms · Sum of n terms of an AP
  $$S_n = \dfrac n2 (a + l) = \dfrac n2 [2a + (n - 1)d]$$
• From a sum formula to the nth term · nth term from the sum
  $$a_n = S_n - S_{n-1}$$
• Symmetric terms and the middle term · Three terms in AP
  $$a - d,\; a,\; a + d \qquad \text{sum} = 3a$$

Watch out for (10)

• n minus 1 steps, not n
  → The nth term of an AP
• Subtract in order
  → The nth term of an AP
• Add the 1
  → Which term is it, and how many terms are there
• A fraction means no
  → Which term is it, and how many terms are there
• Count the steps, not the positions
  → An AP from two of its terms
• Two roots, one answer
  → Sum of the first n terms
• 2a inside the bracket
  → Sum of the first n terms
• S of n is not the nth term
  → From a sum formula to the nth term
• Centre on the middle term
  → Symmetric terms and the middle term
• Both signs of d
  → Symmetric terms and the middle term

Formulas (5)

• The nth term of a GP · nth term of a GP
  $$a_n = a r^{n-1}$$
• Which term is it, and how many terms · Position of a term
  $$r^{n-1} = \dfrac{a_n}{a}$$
• A GP from two of its terms, and the middle term · Ratio from two terms
  $$\dfrac{a_q}{a_p} = r^{\,q-p}$$
• Sum of the first n terms · Sum of n terms of a GP
  $$S_n = \dfrac{a(r^n - 1)}{r - 1}, \quad r \ne 1$$
• Sum to infinity · Sum to infinity
  $$S_\infty = \dfrac{a}{1 - r}, \quad -1 < r < 1$$

Watch out for (9)

• r to the power n minus 1
  → The nth term of a GP
• Bracket a negative ratio
  → The nth term of a GP
• The power plus one
  → Which term is it, and how many terms
• Two square roots
  → A GP from two of its terms, and the middle term
• No term can be zero
  → A GP from two of its terms, and the middle term
• r to the n, not n minus 1
  → Sum of the first n terms
• Subtract 1 before multiplying
  → Sum of the first n terms
• Only when the terms shrink
  → Sum to infinity
• 1 minus r, not r minus 1
  → Sum to infinity

Formulas (3)

• Terms of an HP through the reciprocals · nth term of an HP
  $$h_n = \dfrac{1}{a + (n - 1)d}$$
• The harmonic mean of two numbers · Harmonic mean
  $$H = \dfrac{2ab}{a + b}$$
• AM, GM and HM together, and inserted means · Order of the three means
  $$A \ge G \ge H, \qquad G^2 = AH$$

Reference tables (1)

Watch out for (7)

• Flip back at the end
  → Terms of an HP through the reciprocals
• The terms themselves do not step evenly
  → Terms of an HP through the reciprocals
• Not the average of the reciprocals
  → The harmonic mean of two numbers
• The order matters
  → AP, GP or HP: the middle-term test
• Equal numbers pass every test
  → AP, GP or HP: the middle-term test
• The order is fixed
  → AM, GM and HM together, and inserted means
• G is not the average of A and H
  → AM, GM and HM together, and inserted means

Formulas (1)

• Series built from known pieces · Split and sum
  $$\sum_{k=1}^{n} (u_k + v_k) = \sum_{k=1}^{n} u_k + \sum_{k=1}^{n} v_k$$

Reference tables (1)

Watch out for (4)

• Count the odd numbers
  → The standard sums
• Even numbers are not n squared
  → The standard sums
• Pairs need an even count
  → Series built from known pieces
• A GP of n terms, not an infinite one
  → Series built from known pieces