https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:exponential-vs-linear-growth/v/linear-exponential-models 2023-05-24T15:30:27.32390848Z Exponential vs. linear models: verbal Linear growth occurs at a constant rate, with equal increments added or subtracted over time, while exponential growth involves a constant multiplier that drives an increase or decrease over time. We can look at the type of change over time to see if given example represents linear or exponential growth. Sal Khan video https://cdn.kastatic.org/googleusercontent/tyro7gp_J84Ags1g4nDcm5mrM8SZGsa570Oo0qIbURdnplgjbw3TCDJY9bA69p8NkJIv_4F8fQMyahD5GigieSgJ Exponential vs. linear models: verbal Linear growth occurs at a constant rate, with equal increments added or subtracted over time, while exponential growth involves a constant multiplier that drives an increase or decrease over time. We can look at the type of change over time to see if given example represents linear or exponential growth. https://cdn.kastatic.org/ka-youtube-converted/8OHEgD6YMBw.mp4/8OHEgD6YMBw.mp4 208 Series https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:exponential-vs-linear-growth/v/exponential-vs-linear-growth 2021-02-16T05:02:26Z Exponential vs. linear growth For constant increments in x, a linear growth would increase by a constant difference, and an exponential growth would increase by a constant ratio. video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Exponential vs. linear growth For constant increments in x, a linear growth would increase by a constant difference, and an exponential growth would increase by a constant ratio. https://cdn.kastatic.org/ka-youtube-converted/_vlXdx-CqM0.mp4/_vlXdx-CqM0.mp4 185 Series 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 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 Series 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 Series https://www.khanacademy.org/math/integral-calculus/ic-series/ic-represent-func-as-series/v/writing-series-sigma-notation 2020-09-27T22:59:01Z Converting explicit series terms to summation notation Some sums are really long, but we can use sigma notation to write them in a shorthand way. See such an example here. Sal Khan video https://cdn.kastatic.org/googleusercontent/DnwmdpIJmJV_VSueuSPLMqdAXUgjRZzQmbr1Zm1eJCDYjZuu_fndndfYfUTZV54NX-WVRPaosA2_15NQz44_iMp_ZA Converting explicit series terms to summation notation Some sums are really long, but we can use sigma notation to write them in a shorthand way. See such an example here. https://cdn.kastatic.org/ka-youtube-converted/0DBDuNYsciY.mp4/0DBDuNYsciY.mp4 196 Series https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/v/sigma-notation-sum 2021-06-30T17:32:38Z Summation notation Sigma, Σ, is the standard notation for writing long sums. Learn how it is used in this video. Sal Khan video https://cdn.kastatic.org/googleusercontent/AkUo89aMk-ZXH2KwZTUALeYknZORIY--gS_3ht52xZLOfQSEU8xfeGgm3Ohz0IeLeNCwAh0j_7M_tfBySGHDGA Summation notation Sigma, Σ, is the standard notation for writing long sums. Learn how it is used in this video. https://cdn.kastatic.org/ka-youtube-converted/5jwXThH6fg4.mp4/5jwXThH6fg4.mp4 267 Series https://www.khanacademy.org/math/algebra-home/alg-series-and-induction/alg-advanced-sigma-notation/v/writing-arithmetic-series-in-sigma-notation 2020-06-23T07:22:40Z Arithmetic series in sigma notation Sal writes the arithmetic sum 7+9+11+...+403+405 in sigma notation. There are actually two common ways of doing this. video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Arithmetic series in sigma notation Sal writes the arithmetic sum 7+9+11+...+403+405 in sigma notation. There are actually two common ways of doing this. https://cdn.kastatic.org/ka-youtube-converted/XArCfFaOhoo.mp4/XArCfFaOhoo.mp4 248 Series https://www.khanacademy.org/math/algebra-home/alg-series-and-induction/alg-advanced-sigma-notation/v/writing-geometric-series-in-sigma-notation 2020-10-30T14:35:45Z Finite geometric series in sigma notation Sal writes the geometric sum 2+6+18+54 in sigma notation. There are actually two common ways of doing this. video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Finite geometric series in sigma notation Sal writes the geometric sum 2+6+18+54 in sigma notation. There are actually two common ways of doing this. https://cdn.kastatic.org/ka-youtube-converted/bT_0pn1lHiY.mp4/bT_0pn1lHiY.mp4 194 Series https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:exponential-vs-linear-growth/a/warmup-exponential-vs-linear-growth 2021-04-29T07:59:22Z Warmup: exponential vs. linear growth article 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/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/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:exponential-vs-linear-growth/e/understanding-linear-and-exponential-models 2021-04-29T08:03:48Z Exponential vs. linear models Given a description of a real-world relationship, determine whether that relationship is linear or exponential. Tomer Gal exercise https://www.khanacademy.org/math/algebra-home/alg-series-and-induction/alg-advanced-sigma-notation/e/writing-finite-arithmetic-series-in-sigma-notation 2021-04-29T07:57:38Z Arithmetic series in sigma notation Practice writing finite arithmetic series like 6 + 10 + 14 in sigma notation. Ben Stenhaug exercise https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:series/x9e81a4f98389efdf:geo-series-notation/e/evaluating-basic-sigma-notation 2025-03-01T15:52:37.13987578Z Summation notation intro Evaluate expressions that use sigma notation to make sure you understand what sigma notation means. Ben Stenhaug exercise 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