Sequence and Series: SPI Mathematics

Formulas (4)

• What makes a list an AP · Common difference
  $$d = a_2 - a_1 = a_3 - a_2$$
• The nth term formula · The nth term and the number of terms
  $$a_n = a + (n - 1)d \qquad n = \frac{l - a}{d} + 1$$
• The gap between two terms · Difference of two terms
  $$a_m - a_k = (m - k)d$$
• Terms counted from the end · nth term from the end
  $$l - (n - 1)d$$

Watch out for (10)

• Subtracting the wrong way round
  → What makes a list an AP
• Judging surds by their look
  → What makes a list an AP
• Squares of an AP are not an AP
  → What makes a list an AP
• Using n instead of n minus 1
  → The nth term formula
• Forgetting the +1 when counting terms
  → The nth term formula
• Starting the count at the wrong number
  → The nth term formula
• Counting the terms instead of the steps
  → The gap between two terms
• Losing the sign
  → The gap between two terms
• Getting the direction wrong in a falling AP
  → Terms counted from the end
• Taking n steps back instead of n minus 1
  → Terms counted from the end

Formulas (2)

• The middle term is the average of its neighbours · Three terms in AP
  $$2b = a + c \qquad b = \frac{a + c}{2}$$
• Three unknown terms written around the middle one · Three terms centred on the middle one
  $$(a - d) + a + (a + d) = 3a \qquad (a - d)(a + d) = a^2 - d^2$$

Watch out for (4)

• Writing the middle term as the sum
  → The middle term is the average of its neighbours
• Skipping the check
  → The middle term is the average of its neighbours
• Calling the terms a, a + d, a + 2d
  → Three unknown terms written around the middle one
• Choosing the wrong sign for d
  → Three unknown terms written around the middle one

Formulas (4)

• The two sum formulas · Sum of n terms
  $$S_n = \frac{n}{2}(a + l) = \frac{n}{2}\left[2a + (n - 1)d\right]$$
• Standard sums of natural, odd and even numbers · Standard sums
  $$1 + 2 + \cdots + n = \frac{n(n + 1)}{2} \qquad 1 + 3 + \cdots + (2n - 1) = n^2 \qquad 2 + 4 + \cdots + 2n = n(n + 1)$$
• Finding the number of terms from the sum · Number of terms from the sum
  $$n = \frac{2S}{a + l}$$
• Terms from a sum written in n · Terms from the sum
  $$a_1 = S_1 \qquad a_n = S_n - S_{n-1} \qquad S_n = pn^2 + qn \Rightarrow d = 2p$$

Watch out for (8)

• Giving the last term when the sum is asked
  → The two sum formulas
• Halving twice
  → The two sum formulas
• Mixing up the odd and the even sums
  → Standard sums of natural, odd and even numbers
• Using the last number as n
  → Standard sums of natural, odd and even numbers
• Keeping the negative root
  → Finding the number of terms from the sum
• Forgetting the half
  → Finding the number of terms from the sum
• Reading the sum as a term
  → Terms from a sum written in n
• Expanding the bracket carelessly
  → Terms from a sum written in n

Formulas (2)

• Geometric progression: ratio, nth term and sum · Geometric progression
  $$a_n = ar^{n-1} \qquad S_n = \frac{a(r^n - 1)}{r - 1} \qquad b^2 = ac$$
• Arithmetic and geometric means of two numbers · The two means
  $$\text{AM} = \frac{a + b}{2} \qquad \text{GM} = \sqrt{ab} \qquad t^2 - 2\,\text{AM}\,t + \text{GM}^2 = 0$$

Watch out for (4)

• Using n instead of n minus 1 in the power
  → Geometric progression: ratio, nth term and sum
• Testing a GP by subtracting
  → Geometric progression: ratio, nth term and sum
• Treating the geometric mean as the product
  → Arithmetic and geometric means of two numbers
• Applying a ratio of means to the numbers
  → Arithmetic and geometric means of two numbers