Arithmetic series and partial sums

What a series is

• A series is the sum of the terms of a sequence.
• The partial sum $S_n$ is the total of the first $n$ terms:

$$S_n = u_1 + u_2 + u_3 + \cdots + u_n$$

• Sequence versus series. $4, 7, 10, 13$ is a sequence; $4 + 7 + 10 + 13 = 34$ is the corresponding series. Questions ask for one or the other, and confusing them is a costly misread.

The two formulae

• When you know the first and last terms:

$$S_n = \frac{n}{2}\left(a + u_n\right)$$

• When you know the first term and the common difference:

$$S_n = \frac{n}{2}\left[2a + (n-1)d\right]$$

• Each part:

  • $n$ — the number of terms being added
  • $a$ — the first term
  • $u_n$ — the last term of the sum
  • $d$ — the common difference

• The two formulae are the same thing. Substituting $u_n = a + (n-1)d$ into the first gives the second.

Choosing between them

• Use the first when the question gives you the last term, or when you have already worked it out.
• Use the second when you know $a$, $d$ and $n$ but not the final term — it saves a step.

Why the formula works

• Write the sum forwards and backwards, one under the other:

$$S_n = a + (a+d) + \cdots + (u_n - d) + u_n$$

$$S_n = u_n + (u_n - d) + \cdots + (a+d) + a$$

• Adding the two lines, each column gives the same total $a + u_n$, and there are $n$ columns:

$$2S_n = n\left(a + u_n\right)$$

• Dividing by 2 gives the formula. This is the story of Gauss adding $1$ to $100$ by pairing the ends: $50$ pairs each totalling $101$.

Working through an example

• Find the sum of the first 20 terms of $5, 9, 13, \ldots$
  • $a = 5$, $d = 4$, $n = 20$
  • Using the second formula: $S_{20} = \dfrac{20}{2}\left[2(5) + 19(4)\right] = 10(10 + 76) = 860$
  • Check with the first formula: $u_{20} = 5 + 19(4) = 81$, so $S_{20} = \dfrac{20}{2}(5 + 81) = 10(86) = 860$ ✓

Finding $n$ from a given sum

• Setting $S_n$ equal to a target and solving produces a quadratic in $n$.

• Expand carefully, rearrange to standard form, and solve.

• Reject any negative or fractional root, since $n$ counts terms.

• Worked through — how many terms of $3, 7, 11, \ldots$ sum to 210?

  • $$\dfrac{n}{2}\left[6 + (n-1)4\right] = 210$$
  • $\dfrac{n}{2}(4n + 2) = 210$, so $n(2n+1) = 210$
  • $2n^2 + n - 210 = 0$, giving $(2n + 21)(n - 10) = 0$
  • $n = 10$ or $n = -10.5$; reject the negative, so $n = 10$

Sums in context

• The give-away is a total accumulated over a number of equal periods:

  • The total saved after 12 months of increasing deposits
  • The total distance covered by an athlete adding a fixed extra distance each week
  • The total number of seats in a theatre with rows increasing steadily

• "How much in the 10th month" is a sequence question ($u_{10}$); "how much altogether after 10 months" is a series question ($S_{10}$). Read which is wanted.