Unit 10: Sequences and Series
Lists of numbers built by a fixed rule, and what happens when you start adding every term in the list together.
7 topics · 26 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.
Arithmetic Sequences
- How it worksHow does the common difference let you jump straight to the 50th term without listing the other 49?
- Worked exampleFind the 20th term of the arithmetic sequence 7, 11, 15, 19.
- Common mistakeWhy isn't every sequence that goes up by the same amount automatically written starting from term zero?
Your turn1 of 2
Find the nth term of the arithmetic sequence with first term 7 and common difference 6, as a function of n.Arithmetic Series
- How it worksHow does Gauss's pairing trick turn into the formula for the sum of an arithmetic series?
- Worked exampleFind the sum of the first 30 terms of the arithmetic sequence 4, 9, 14, 19.
- Common mistakeWhy can't you find the sum of an arithmetic series just by multiplying two random terms by the count?
- Ties togetherHow does the arithmetic series formula grow out of the same common difference used in the arithmetic sequence formula?
Your turn1 of 2
Find 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
- How it worksHow does the common ratio let you jump straight to any term of a geometric sequence using an exponent?
- Worked exampleFind the 8th term of the geometric sequence 3, 6, 12, 24.
- Common mistakeWhy doesn't a geometric sequence with a ratio between 0 and 1 count as growing?
Your turn1 of 2
Find the nth term of the geometric sequence with first term 7 and common ratio 3, as a function of n.Finite Geometric Series
- How it worksHow does multiplying a geometric series by its own ratio and subtracting cancel almost every term?
- Worked exampleFind the sum of the first 10 terms of the geometric sequence 2, 6, 18, 54.
- Common mistakeWhy can't the sum of a geometric series be found by just multiplying the first term by the number of terms?
- Going furtherWhat happens to the geometric series sum formula's derivation when the common ratio equals exactly 1?
Your turn1 of 2
Find the sum of the first n terms of the geometric sequence with first term 7 and common ratio 3, as a function of n.Infinite Geometric Series
- How it worksHow does a common ratio smaller than 1 in size shrink each new term enough for the total to settle down?
- Worked exampleFind the sum of the infinite geometric series 8 + 4 + 2 + 1 + ...
- Common mistakeWhy does an infinite geometric series with a ratio bigger than 1 refuse to add up to any finite number?
- Going furtherWhat happens to the sum of an infinite geometric series as its ratio gets closer and closer to 1?
- Ties togetherHow does the infinite geometric series formula explain why a repeating decimal like 0.333... equals exactly one third?
Your turn1 of 2
Find the sum of the infinite geometric series with first term 14 and common ratio 1/7 (where 7 is greater than 1, so the ratio is between 0 and 1).Sigma Notation for Sums
- How it worksHow do the top and bottom numbers on a sigma tell you exactly which terms to add?
- Worked exampleFind the sum of 3k + 1 for k from 1 to 6 using sigma notation.
- Common mistakeWhy doesn't changing the index letter in sigma notation, like from k to n, change the value of the sum?
Your turn1 of 2
Find the sum of 7k + 6 for k from 1 to n, using sigma notation, as a function of n.Recursive vs. Explicit Formulas for Sequences
- How it worksHow do you convert a recursive definition of a sequence into a direct, explicit formula?
- Worked exampleFind the explicit formula for the sequence defined recursively by a(1) = 5 and a(n) = a(n-1) + 3.
- Common mistakeWhy can't you find the 100th term of a recursive sequence by just plugging 100 into the recursive rule?
- Ties togetherHow does converting a recursive rule into an explicit formula here mirror finding the explicit formula for a geometric sequence?
Your turn1 of 2
A sequence is defined recursively by a(1) = 7 and a(n) = a(n-1) + 6. Find the explicit formula for a(n), as a function of n.