https://www.khanacademy.org/computing/pixar/pattern/dino/v/creating-patterns-pixar-in-a-box 2023-07-31T16:40:28.830178554Z Introduction to Patterns Pixar shading artists Beth and Ana explain how patterns play a crucial role in creating believable textures for animated characters like Arlo from The Good Dinosaur. They discuss color, illumination, and displacement, and emphasize the importance of adding randomness to patterns for a more organic and realistic appearance. video https://cdn.kastatic.org/googleusercontent/_lpxi903ws5XHrgacKh2R-OClX9v9MfhVJLqD6D5N-3EIeaL4aueZlF0CemXNIBQmlygiQ6Oci0bnW1ib1jUsOlu Introduction to Patterns Pixar shading artists Beth and Ana explain how patterns play a crucial role in creating believable textures for animated characters like Arlo from The Good Dinosaur. They discuss color, illumination, and displacement, and emphasize the importance of adding randomness to patterns for a more organic and realistic appearance. https://cdn.kastatic.org/ka-youtube-converted/YFzhoXmbhtI.mp4/YFzhoXmbhtI.mp4 163 Patterns, sequences and series https://www.khanacademy.org/math/class-6-ncf/x04ca845a9a148c80:patterns-in-mathematics-india/x04ca845a9a148c80:visualising-number-sequences-india/v/math-patterns-toothpicks-indian-accent 2025-05-29T05:05:30.332110463Z Math patterns: toothpicks Math patterns: toothpicks Skyloom (Dubbing) video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Math patterns: toothpicks Math patterns: toothpicks https://cdn.kastatic.org/ka-youtube-converted/XUonAWQermE.mp4/XUonAWQermE.mp4 285 Patterns, sequences and series https://www.khanacademy.org/math/uk-math-class-6/x2779fb3c4322516d:algebra-ncert-new/x2779fb3c4322516d:math-patterns/v/math-patterns-table 2025-06-25T05:51:35.728021887Z Math patterns: table We explain a pattern with the number of seats at a table. Skyloom (Dubbing) video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Math patterns: table We explain a pattern with the number of seats at a table. https://cdn.kastatic.org/ka-youtube-converted/Rp_XIuxF5Mk.mp4/Rp_XIuxF5Mk.mp4 204 Patterns, sequences and series https://www.khanacademy.org/math/cc-fourth-grade-math/imp-factors-multiples-and-patterns/imp-math-patterns/e/patterns-with-shapes 2026-04-30T20:20:55.724408584Z Patterns with shapes Practice identifying features of a pattern with shapes. Kami Soltesz exercise https://www.khanacademy.org/math/class-6-ncf/x04ca845a9a148c80:patterns-in-mathematics-india/x04ca845a9a148c80:patterns-in-shapes-india/v/patterns-in-shapes 2025-12-05T09:40:41.786169848Z Patterns in Shapes In this video, we will discuss some interesting shape patterns (and fractals). Khan Academy video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Patterns in Shapes In this video, we will discuss some interesting shape patterns (and fractals). https://cdn.kastatic.org/ka-youtube-converted/D1KWce8Sbkg.mp4/D1KWce8Sbkg.mp4 310 Patterns, sequences and series https://www.khanacademy.org/math/class-6-ncf/x04ca845a9a148c80:patterns-in-mathematics-india/x04ca845a9a148c80:visualising-number-sequences-india/v/fibonacci-visualisation 2026-03-25T17:10:42.46906482Z Fibonacci Visualisation In this video, we will see what is a Fibonacci sequence, how to visualise them and what are hexagonal numbers. Khan Academy video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Fibonacci Visualisation In this video, we will see what is a Fibonacci sequence, how to visualise them and what are hexagonal numbers. https://cdn.kastatic.org/ka-youtube-converted/MTf90nyB55Y.mp4/MTf90nyB55Y.mp4 367 Patterns, sequences and series https://www.khanacademy.org/college-careers-more/bjc/x4c51c62a:2022-challenge/x4c51c62a:2022-challenge-popular-vote-regional-champions/v/josefina-chile-math-s-america-region-champion-semifinalist-2022-breakthrough-junior-challenge 2023-04-06T04:51:06.745361174Z Fibonacci Numbers FIBONACCI NUMBERS Learn more at https://breakthroughjuniorchallenge.org. The Breakthrough Junior Challenge is an annual competition for students, ages 13-18, to share their passion for math and science with the world! Each student submits a video that explains a challenging and important concept or theory in mathematics, life sciences, or physics. The winner receives a $250,000 college scholarship. The winning student's teacher and school also benefit: $50,000 for the teacher and a state-of-the-art $100,000 science lab for the school. Khan Academy video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Fibonacci Numbers FIBONACCI NUMBERS Learn more at https://breakthroughjuniorchallenge.org. The Breakthrough Junior Challenge is an annual competition for students, ages 13-18, to share their passion for math and science with the world! Each student submits a video that explains a challenging and important concept or theory in mathematics, life sciences, or physics. The winner receives a $250,000 college scholarship. The winning student's teacher and school also benefit: $50,000 for the teacher and a state-of-the-art $100,000 science lab for the school. https://cdn.kastatic.org/ka-youtube-converted/GzM_zfDHajM.mp4/GzM_zfDHajM.mp4 90 Patterns, sequences and series https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:sequences-and-series-ncert-new/x0419e5b3b578592a:series/v/worked-example-fibonacci-sequence 2025-03-01T15:52:37.13987578Z Worked example - Fibonacci sequence Given the recursion formula for Fibonacci sequence, how do we generate the list of first few numbers. We first try playing with the recursion formula but then quickly realise the pattern. Every term is equal to the sum of the previous two. The question asks us to find the ratio of consecutive terms which means we briefly get to say hello to the golden ratio! Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Worked example - Fibonacci sequence Given the recursion formula for Fibonacci sequence, how do we generate the list of first few numbers. We first try playing with the recursion formula but then quickly realise the pattern. Every term is equal to the sum of the previous two. The question asks us to find the ratio of consecutive terms which means we briefly get to say hello to the golden ratio! https://cdn.kastatic.org/ka-youtube-converted/KaF0rEG8ySM.mp4/KaF0rEG8ySM.mp4 346 Patterns, sequences and series https://www.khanacademy.org/math/class-6-ncf/x04ca845a9a148c80:patterns-in-mathematics-india/x04ca845a9a148c80:patterns-in-numbers-india/v/recognizing-number-patterns-indian-accent 2025-05-29T05:05:30.332110463Z Recognizing number patterns Recognizing number patterns (Indian Accent) Aakash Bagga video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Recognizing number patterns Recognizing number patterns (Indian Accent) https://cdn.kastatic.org/ka-youtube-converted/6sJLCiWtQAc.mp4/6sJLCiWtQAc.mp4 147 Patterns, sequences and series https://www.khanacademy.org/math/cc-fourth-grade-math/imp-factors-multiples-and-patterns/imp-math-patterns/e/math-patterns 2026-07-01T14:50:47.194464438Z Patterns with numbers Generate terms in a pattern when given a rule.  Identify features of a pattern that are not explicit to the rule itself. Tomer Gal exercise https://www.khanacademy.org/math/cc-fifth-grade-math/imp-algebraic-thinking/imp-number-patterns/e/visualizing-and-interpreting-relationships-between-patterns 2026-03-06T17:10:44.240025138Z Relationships between 2 patterns Generate patterns using given rules. Identify relationships between terms. Graph ordered pairs consisting of corresponding terms from the patterns. lisa exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:introduction-to-arithmetic-sequences/v/explicit-and-recursive-definitions-of-sequences 2021-06-30T17:32:35Z Sequences intro Sequences are ordered lists of numbers (called "terms"), like 2,5,8. Some sequences follow a specific pattern that can be used to extend them indefinitely. For example, 2,5,8 follows the pattern "add 3," and now we can continue the sequence. Sequences can have formulas that tell us how to find any term in the sequence. For example, 2,5,8,... can be represented by the formula 2+3(n-1). Sal Khan video https://cdn.kastatic.org/googleusercontent/hVR9ztsfKuZmzNpTPAxmSv9yKwniky9Ud8fk0H9PpjTNOPP7n23hO9EZYV0S4oxKAceRFSgnRrOzuVCTqIW3dJVP Sequences intro Sequences are ordered lists of numbers (called "terms"), like 2,5,8. Some sequences follow a specific pattern that can be used to extend them indefinitely. For example, 2,5,8 follows the pattern "add 3," and now we can continue the sequence. Sequences can have formulas that tell us how to find any term in the sequence. For example, 2,5,8,... can be represented by the formula 2+3(n-1). https://cdn.kastatic.org/ka-youtube-converted/KRFiAlo7t1E.mp4/KRFiAlo7t1E.mp4 498 Patterns, sequences and series https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:introduction-to-arithmetic-sequences/v/arithmetic-sequences 2021-06-30T17:32:39Z Intro to arithmetic sequences Sal introduces arithmetic sequences and their main features, the initial term and the common difference. He gives various examples of such sequences, defined explicitly and recursively. Sal Khan video https://cdn.kastatic.org/googleusercontent/OrgsHeMoJUc6-j5RLRoUIVcaYF-Q39nGk23bLLQKOPj4-e45JJZypOWsNNXFilJ9aITPS6AX4kJKgtfWn7HFcAlI Intro to arithmetic sequences Sal introduces arithmetic sequences and their main features, the initial term and the common difference. He gives various examples of such sequences, defined explicitly and recursively. https://cdn.kastatic.org/ka-youtube-converted/_cooC3yG_p0.mp4/_cooC3yG_p0.mp4 427 Patterns, sequences and series https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:introduction-to-arithmetic-sequences/a/introduction-to-arithmetic-sequences 2021-07-02T22:45:33Z Intro to arithmetic sequences Get comfortable with sequences in general, and learn what arithmetic sequences are. article https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:constructing-arithmetic-sequences/a/arithmetic-sequences-review 2021-04-29T07:49:07Z Arithmetic sequences review Review arithmetic sequences and solve various problems involving them. article https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:introduction-to-arithmetic-sequences/e/arithmetic_sequences_1 2025-12-30T07:05:15.972746521Z Extend arithmetic sequences Given the first few terms in an arithmetic sequence, find the next term in the sequence. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:introduction-to-geometric-sequences/v/geometric-sequences-introduction 2021-06-30T17:32:40Z Intro to geometric sequences Sal introduces geometric sequences and their main features, the initial term and the common ratio. Sal Khan video https://cdn.kastatic.org/googleusercontent/EGwJyoWiNK9S9o40xAqCEN70zhYCxmMpzmSCp5vp890CX6dQLAk-NetWrz9x17rxaUPCwwEGRpjxhQT_2WdqYjk Intro to geometric sequences Sal introduces geometric sequences and their main features, the initial term and the common ratio. https://cdn.kastatic.org/ka-youtube-converted/pXo0bG4iAyg.mp4/pXo0bG4iAyg.mp4 645 Patterns, sequences and series https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:introduction-to-geometric-sequences/e/geometric_sequences_1 2025-11-05T21:00:42.500191293Z Extend geometric sequences Given the first few terms in a geometric sequence, find the next term in the sequence. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:modeling-with-sequences/v/modeling-situations-with-arithmetic-and-geometric-sequences 2022-10-06T23:01:00.745424633Z Sequences word problems Sal solves two word problems about modeling real-world situations with arithmetic and geometric sequences. video https://cdn.kastatic.org/ka_thumbnails_cache/9c74164f-e41d-44fd-9987-9c6d7c819eba_1280_720_base.png Sequences word problems Sal solves two word problems about modeling real-world situations with arithmetic and geometric sequences. https://cdn.kastatic.org/ka-youtube-converted/yYGf7xn7TyM.mp4/yYGf7xn7TyM.mp4 343 Patterns, sequences and series https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:modeling-with-sequences/e/recursive_explicit 2026-04-08T00:16:02.937952483Z Sequences word problems Given a verbal description of a real-world relationship, determine the sequence that models that relationship. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:constructing-arithmetic-sequences/v/finding-the-100th-term-in-a-sequence 2021-02-16T05:02:27Z Arithmetic sequence problem Sal finds the 100th term in the sequence 15, 9, 3, -3... Sal Khan video https://cdn.kastatic.org/googleusercontent/x3JWFXznBvlgzB1Y7p3RUF9raYYEkKlf4_Rnl7dnFR4XAfw0nNAF0dkNJhaO63sKgmGsoqapSDiOfDNs4uvvVb8m Arithmetic sequence problem Sal finds the 100th term in the sequence 15, 9, 3, -3... https://cdn.kastatic.org/ka-youtube-converted/C5titprQAc4.mp4/C5titprQAc4.mp4 357 Patterns, sequences and series https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:series/x9e81a4f98389efdf:arith-series/a/series-faq 2022-10-26T00:13:38.755031109Z Series: FAQ Frequently asked questions on series article https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:series/x9e81a4f98389efdf:arith-series/v/sum-of-arithmetic-sequence-arithmetic-series 2022-10-07T00:22:18.752150461Z Arithmetic series intro Sal explains the formula for the sum of a finite arithmetic series Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/85b3e32a-0fa7-4f0d-950b-7ae0a2a5b525_1280_720_base.png Arithmetic series intro Sal explains the formula for the sum of a finite arithmetic series https://cdn.kastatic.org/ka-youtube-converted/cYw4MFWsB6c.mp4/cYw4MFWsB6c.mp4 236 Patterns, sequences and series https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:sequences-and-series-ncert-new/x0419e5b3b578592a:series/v/arithmetic-series-formula-indian-accent 2025-06-03T19:25:43.416794683Z Arithmetic series formula Learn how to write an arithmetic sequence in general terms, using a common difference d and a first term a. Next, we'll see that this formula is equivalent to multiplying the average of the first and last terms by the number of terms. Kalakrit (Dubbing) video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Arithmetic series formula Learn how to write an arithmetic sequence in general terms, using a common difference d and a first term a. Next, we'll see that this formula is equivalent to multiplying the average of the first and last terms by the number of terms. https://cdn.kastatic.org/ka-youtube-converted/o8NvEAmqCUs.mp4/o8NvEAmqCUs.mp4 466 Patterns, sequences and series https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:series/x9e81a4f98389efdf:arith-series/a/evaluating-arithmetic-series-guided-practice 2021-06-30T17:32:39Z Arithmetic series Walk through a guided practice where you'll start by finding a simple sum and end by evaluating finite arithmetic series. article https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:series/x9e81a4f98389efdf:arith-series/a/evaluating-arithmetic-series-worksheet 2021-06-30T17:32:36Z Arithmetic series worksheet Try a few problems where you'll be asked to evaluate finite arithmetic series. article https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:series/x9e81a4f98389efdf:arith-series/e/arithmetic_series 2025-03-01T15:52:37.13987578Z Arithmetic series Practice evaluating arithmetic series using the formula (n/2)⋅(a₁+aₙ). Ben Alpert exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-factor/x2ec2f6f830c9fb89:geo-series/v/geo-series-intro 2023-09-22T17:43:53.269744476Z Geometric series introduction Learn how money grows in a bank account with geometric series! Discover how each deposit grows by a fixed percentage every year, creating a pattern. This pattern forms a geometric series, a useful concept in finance and business. Keep depositing and watch your money multiply! video https://cdn.kastatic.org/googleusercontent/Gz4AOvHeDFajnCPR_61R9pMjlALRpHI0ySpVwNI88eTAqSJQWBDus9N44HIarGzOzwIdJ_lBcL3fV0USwNPJPoI Geometric series introduction Learn how money grows in a bank account with geometric series! Discover how each deposit grows by a fixed percentage every year, creating a pattern. This pattern forms a geometric series, a useful concept in finance and business. Keep depositing and watch your money multiply! https://cdn.kastatic.org/ka-youtube-converted/BdsURIUQ04k.mp4/BdsURIUQ04k.mp4 377 Patterns, sequences and series https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:series/x9e81a4f98389efdf:geo-series/v/series-as-sum-of-sequence 2020-11-10T16:09:48Z Geometric series intro A geometric series is the sum of the terms of a geometric sequence. Learn more about it here. Sal Khan video https://cdn.kastatic.org/googleusercontent/TmSdYcKy73Yzh2Lwlb3RhiRliE_Ippk6x9DIoAtEk2197evzFXj08U7DSr91qZTj072cpqND7DIf_hRnrG9-NjwG Geometric series intro A geometric series is the sum of the terms of a geometric sequence. Learn more about it here. https://cdn.kastatic.org/ka-youtube-converted/rcRg_gO7-7E.mp4/rcRg_gO7-7E.mp4 173 Patterns, sequences and series https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-factor/x2ec2f6f830c9fb89:geo-series/v/deriving-formula-for-sum-of-finite-geometric-series 2023-09-22T17:43:53.269744476Z Finite geometric series formula A finite geometric series can be solved using the formula a(1-rⁿ)/(1-r). Sal demonstrates how to derive a formula for the sum of the first 'n' terms of such a series, emphasizing the importance of understanding the number of terms being summed. video https://cdn.kastatic.org/ka_thumbnails_cache/d2be81fa-a733-4db8-8f50-05e253e9e679_1280_720_base.png Finite geometric series formula A finite geometric series can be solved using the formula a(1-rⁿ)/(1-r). Sal demonstrates how to derive a formula for the sum of the first 'n' terms of such a series, emphasizing the importance of understanding the number of terms being summed. https://cdn.kastatic.org/ka-youtube-converted/Cf0-6ngH2gs.mp4/Cf0-6ngH2gs.mp4 435 Patterns, sequences and series https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-factor/x2ec2f6f830c9fb89:geo-series/v/three-examples-of-evaluating-finite-geometric-series 2023-09-22T17:55:38.713124768Z Worked examples: finite geometric series Get a front row seat to Sal's step-by-step solutions to finite geometric series. Learn how to find the sum of the first 50 terms of a series by multiplying each term by a common ratio. Discover how to apply the formula for the sum of a finite geometric series. Plus, explore how to handle series with a change in sign. video https://cdn.kastatic.org/ka_thumbnails_cache/7326ea77-7c8a-4068-acaa-8406f4504c29_1280_720_base.png Worked examples: finite geometric series Get a front row seat to Sal's step-by-step solutions to finite geometric series. Learn how to find the sum of the first 50 terms of a series by multiplying each term by a common ratio. Discover how to apply the formula for the sum of a finite geometric series. Plus, explore how to handle series with a change in sign. https://cdn.kastatic.org/ka-youtube-converted/DY9Q3qNmZnw.mp4/DY9Q3qNmZnw.mp4 415 Patterns, sequences and series https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-factor/x2ec2f6f830c9fb89:geo-series/e/geometric-series 2025-03-01T15:52:37.13987578Z Finite geometric series word problems Understanding and solving problems with the formula for a finite geometric series Rhonda Hughes exercise