Eleventh Grade · Math · week 30

Arithmetic sequences and series

Find terms and partial sums of arithmetic sequences and series using explicit formulas.

An arithmetic sequence adds the same fixed number -- the common difference, d -- to get from one term to the next. Its explicit formula jumps straight to any term without listing every one before it: a_n = a_1 + (n-1)d, where a_1 is the first term and n counts which term you want. The (n-1) matters: the first term has taken zero "hops" of size d, the second term has taken one hop, and so on.

An arithmetic SERIES is just the sum of a sequence's terms, and it has a beautifully simple formula: S_n = (n/2)(a_1 + a_n) -- pair the first and last terms, the second and second-to-last, and so on; every pair adds up to the same total (a_1+a_n), and there are n/2 such pairs. This is the same trick a young Gauss reportedly used to sum 1 through 100 in seconds.

📋 For the teacher
Teach it (10–15 min):
  • Have them physically count "hops" on their fingers for a small n (like n=4) so the (n-1), not n, in the formula stops feeling arbitrary.
  • Tell the (probably embellished, but instructive) story of young Gauss summing 1 to 100 by pairing -- it makes the sum formula feel discovered, not just handed down.
  • For a word problem like the theater, help them identify a_1 and d from the story FIRST, explicitly, before reaching for either formula.
  • Practice both formulas as two separate, sequential steps for word problems: find the needed term first, THEN find the sum -- do not try to skip straight to a sum without knowing a_n.
Talk about it:
  • Why is it (n-1) hops in the explicit formula, and not n hops?
  • How does pairing the first and last terms explain the sum formula?
  • How do you find a_1 and d from a word problem that does not use that vocabulary?

Mastery looks like: They find any term of an arithmetic sequence with the explicit formula and the sum of a specified number of terms with the series formula.

Common stumbles: Using n instead of (n-1) in the explicit formula; trying to find a sum without first finding the needed last term.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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