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Unit 14: Sequences, Series, and Mathematical Induction

Patterns built term by term, the sums of arithmetic and geometric lists, proving a statement true for every integer at once, and expanding a binomial.

9 topics · 35 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Sequences and Recursive Formulas

  1. How it worksHow does a recursive formula generate an entire sequence from just one starting value and a rule?
  2. Worked exampleFind the first five terms of the sequence defined by a1 = 3 and a(n) = 2·a(n-1) + 1.
  3. Common mistakeWhy isn't a recursive formula the same thing as an explicit formula for the nth term?
  4. Ties togetherHow does a recursive rule with a constant added difference turn into the arithmetic sequences covered next?
Your turn1 of 2
A sequence has a1 = 7 and a(n) = 3·a(n-1) + 8. Find a3.

Arithmetic Sequences

  1. How it worksHow does a constant common difference build the explicit formula a(n) = a1 + (n-1)d?
  2. Worked exampleFind the 20th term of the arithmetic sequence 5, 9, 13, 17, ...
  3. Common mistakeWhy isn't every sequence that increases by roughly the same amount actually arithmetic?
Your turn1 of 2
Write the nth term of the arithmetic sequence with first term 7 and common difference 6, as a function of n.

Arithmetic Series

  1. How it worksHow does pairing the first and last terms of an arithmetic series build the formula Sn = n(a1+an)/2?
  2. Worked exampleFind the sum of the first 30 terms of the arithmetic sequence 4, 10, 16, 22, ...
  3. Worked exampleFind the sum of the arithmetic series 3 + 7 + 11 + ... + 79.
  4. Common mistakeWhy can't you find the sum of a series just by multiplying a guessed average term by the number of terms without checking it's arithmetic?
Your turn1 of 2
Write the sum of the first n terms of the arithmetic sequence with first term 7 and common difference 6, as a function of n.

Geometric Sequences

  1. How it worksHow does a constant common ratio build the explicit formula a(n) = a1·rn-1?
  2. Worked exampleFind the 8th term of the geometric sequence 2, 6, 18, 54, ...
  3. Common mistakeWhy doesn't a geometric sequence with a ratio between 0 and 1 count as decreasing forever without limit?
Your turn1 of 2
Write the nth term of the geometric sequence with first term 7 and common ratio 2.47, as a function of n.

Geometric Series

  1. How it worksHow does multiplying a geometric series by its ratio and subtracting produce the finite sum formula?
  2. Worked exampleFind the sum of the first 10 terms of the geometric series 3 + 6 + 12 + 24 + ...
  3. Common mistakeWhy doesn't the formula for a finite geometric series work when the common ratio equals exactly 1?
  4. Ties togetherHow does the finite geometric series formula change once the number of terms grows without bound?
Your turn1 of 2
Write the sum of the first n terms of the geometric series with first term 7 and common ratio 2.47, as a function of n.

Infinite Geometric Series

  1. How it worksHow does the formula S = a1/(1-r) come from letting the number of terms in a geometric series grow without bound?
  2. Worked exampleFind the sum of the infinite geometric series 8 + 4 + 2 + 1 + ...
  3. Common mistakeWhy doesn't an infinite geometric series have a finite sum when the common ratio's absolute value is greater than 1?
  4. Common mistakeWhy doesn't an infinite geometric series converge when its ratio is exactly 1 or exactly -1?
Your turn1 of 2
Find the sum of the infinite geometric series with first term 7 and common ratio 0.27, where |0.27| < 1.

Mathematical Induction

  1. How it worksHow do the base case and the inductive step together prove a statement true for every natural number?
  2. Worked exampleProve by induction that 1 + 2 + 3 + ... + n equals n(n+1)/2 for every positive integer n.
  3. Worked exampleProve by induction that 2n is greater than n for every positive integer n.
  4. Common mistakeWhy doesn't checking a formula for the first ten values of n count as a proof by induction?
  5. Going furtherWhat happens to an induction proof's base case when the statement is only claimed to be true starting at n = 5?
Your turn1 of 2
Write the closed-form value of 1 + 2 + 3 + ... + n (the claim proven true for every n by induction), as a function of n.

The Binomial Theorem

  1. How it worksHow does the Binomial Theorem use combinations to generate each term's coefficient in an expansion?
  2. Worked exampleFind the expansion of (x+2)4 using the Binomial Theorem.
  3. Worked exampleFind the coefficient of the x3·y5 term in the expansion of (x+y)8.
  4. Common mistakeWhy isn't (x+y)2 equal to x2 + y2?
Your turn1 of 2
Write the coefficient of the x2·yn-2 term in the expansion of (x+y)n, as a function of n.

Pascal's Triangle and Binomial Coefficients

  1. How it worksHow does each entry of Pascal's triangle come from adding the two entries diagonally above it?
  2. Worked exampleFind the 5th row of Pascal's triangle and use it to expand (x+y)4.
  3. Common mistakeWhy isn't the binomial coefficient n choose k the same thing as n divided by k?
  4. Ties togetherHow does each row of Pascal's triangle line up exactly with the coefficients the Binomial Theorem predicts?
Your turn1 of 2
Write the value of n choose 3 (the 4th entry of row n in Pascal's triangle), as a function of n.