Finding the nth term of a Quadratic Sequence - GCSE Higher Maths

A video explaining how to find the nth term of a quadratic sequence. This topic is on the GCSE Maths Higher course and not foundation. If you want to check linear sequences use this video: • Sequences and Finding ...
Exam Question Booklets:📝
🔗Exam Question Edexcel Style: www.1stclassmaths.com/_files/...
🔗Exam Questions AQA Style: www.1stclassmaths.com/_files/...
🌐 www.1stclassmaths.com/ for more high quality revision questions.
0:00 Intro
0:18 Finding the second difference
1:26 Finding the coefficient of n squared
1:58 Finding the linear part
3:23 Finding the nth term of the quadratic sequence
3:48 A second example
To help my channel:
❤️Like
💬Comment
🔔Subscribe
Follow me on:
/ 1stclassmaths
/ 1stclassmaths
/ 1stclassmaths

Пікірлер: 179

  • @Ramimhasan-rd8oc
    @Ramimhasan-rd8oc5 ай бұрын

    this is the most clear explanation i have ever watched.. Thanks a lot

  • @1stClassMaths

    @1stClassMaths

    5 ай бұрын

    Glad to hear that!

  • @RobloxianX
    @RobloxianX3 ай бұрын

    Thank you so much, my school did a 1 hour lesson on this and they werent able to teach anyone. I sent this to everyone in my class, we are saved! Thank you so much, you're doing gods work

  • @JakeOfShire
    @JakeOfShire5 ай бұрын

    Wow! this was honestly life changing. It makes a thousand times more sense. I no longer look at this question with dread! Thank you 😅

  • @Storm_Fly
    @Storm_Fly2 ай бұрын

    By far the most clear explanation and straight to the point . Thank u sir !

  • @staroffixial8159
    @staroffixial81597 ай бұрын

    Thank you so much! Now im prepared for my gcses :)

  • @Stratmaster_
    @Stratmaster_8 ай бұрын

    You have no idea how much this video helps before an exam!!! 😂 🫡

  • @Goose_G.O.D

    @Goose_G.O.D

    3 ай бұрын

    😂😂 facts

  • @danielippolito-fletcherscr3273
    @danielippolito-fletcherscr32735 ай бұрын

    This is a spectacular explanation. I am a middle school math teacher who has been struggling to grasp an easy way to explain this for weeks. Thanks so much!

  • @1stClassMaths

    @1stClassMaths

    5 ай бұрын

    Excellent I am glad it was so useful! I picked it up off another teacher a few years ago :)

  • @Lafentity
    @Lafentity6 ай бұрын

    Bro thanks so much. This deserves a sub, you teach a much clearer explanation than my tutor.

  • @user-wn5yi3pc5s
    @user-wn5yi3pc5s4 ай бұрын

    Amazing explanation!

  • @z.a075
    @z.a07528 күн бұрын

    this is honestly so so helpful. may god bless you. you have just saved me

  • @1stClassMaths

    @1stClassMaths

    27 күн бұрын

    You're so welcome!

  • @pluto_2682
    @pluto_26823 ай бұрын

    You just saved me 😭 literally writing my math exam tomorrow .. great video

  • @theokongnaphakdee9391
    @theokongnaphakdee9391 Жыл бұрын

    thank you so much!!!! that was very helpful!!!!!!🙏🙏🙏🙏🙏

  • @alisa6170
    @alisa61705 ай бұрын

    the explanation and visuals were so easy to understand!! thank you so much

  • @1stClassMaths

    @1stClassMaths

    5 ай бұрын

    You're very welcome!

  • @yoitsfaizy8778
    @yoitsfaizy8778Ай бұрын

    very comprehensible, cheers m8

  • @user-fg2cn7si6u
    @user-fg2cn7si6uАй бұрын

    Thank u for the nice explanation. I finally undertood this!!

  • @1stClassMaths

    @1stClassMaths

    Ай бұрын

    You are welcome!

  • @GamingWithJas-ff4yf
    @GamingWithJas-ff4yfАй бұрын

    You taught this much better than my teacher, my teacher explained it differently, i did understand but your explanation is way better, thank you a lot im using you to revise for my end of year 9 assessment

  • @Fire.edits.14
    @Fire.edits.144 ай бұрын

    thank you so much, i tried watching a lot of videos, this is the best and most clear one :)

  • @1stClassMaths

    @1stClassMaths

    4 ай бұрын

    Glad to hear that!

  • @user-hs1pt7ud7r
    @user-hs1pt7ud7r4 ай бұрын

    Hello teacher so thank you because your teaching so simply God willing every time get happy ❤

  • @gracehathaway2715
    @gracehathaway27156 ай бұрын

    Honestly, ty so much .I tried to watch the corbet maths video on this and was on the verge of tears 😂.this vid makes so much sense now ❤

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    You're very welcome!

  • @truthwins5804
    @truthwins58046 ай бұрын

    It helped a lot . Thanks ❤❤❤ I didn’t understand a thing about how to find the nth term of a quadratic sequence but after this video , you would think I was Einstein!! 😂😂 Thanks again😊😊

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    You are welcome….Einstein. 😁

  • @tatertotsassemble
    @tatertotsassemble2 ай бұрын

    this was so clear! my teacher has been so difficult with explaining it to me lately, and always got annoyed when i asked “too many questions” so i haven’t been able to understand ANY of it. but this cleared pretty much everything up. as a student in year 9, i feel much more prepared for my gcse prep now

  • @ieatmentoss1714

    @ieatmentoss1714

    2 ай бұрын

    you should get another teacher if they're saying you ask “too many questions”

  • @RedditsFunTales

    @RedditsFunTales

    2 ай бұрын

    @@ieatmentoss1714 exactly, school is a place for development. teachers should be pleased if a student asks a lot of questions as it shows their interest in the topic and forecasts they need help. must be a crappy teacher if that is their excuse.

  • @PKisBetter

    @PKisBetter

    2 ай бұрын

    bro ur doing this in 9th grade?

  • @kanan597

    @kanan597

    2 ай бұрын

    youre in y9 theres no need to be prepping for gcses rn, take it from a y11 :)

  • @aleynak-wn5xq

    @aleynak-wn5xq

    2 ай бұрын

    @@kanan597they still have mocks and internal exams

  • @ElaineWilkins-v1k
    @ElaineWilkins-v1k25 күн бұрын

    You are a legend thank you for explaining it so well

  • @1stClassMaths

    @1stClassMaths

    21 күн бұрын

    Glad it helped!

  • @MuhammadHammad-k7i
    @MuhammadHammad-k7i10 күн бұрын

    best explination ever sir thanks a lot

  • @charles-np1rc
    @charles-np1rc3 ай бұрын

    thank you helped a lot

  • @user-es1pk4og1z
    @user-es1pk4og1z4 ай бұрын

    Thank you very much!

  • @Crukks1973
    @Crukks19735 ай бұрын

    how do you get the values 1,4,9,16, and 25 as the square numbers

  • @1stClassMaths

    @1stClassMaths

    5 ай бұрын

    1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 5 x 5 = 25

  • @user-dm4sx6kt8k
    @user-dm4sx6kt8k10 ай бұрын

    Thanks!

  • @thesingingchef29
    @thesingingchef29Ай бұрын

    Thank you so much! Also here's another method: a: second diff / 2 b: 3a+b = first diff of first term c: a+b+c = first term Now, all you have to do is sub a, b, and c into the equation T(n) = an²+bn+c. *Leave T and n as variables

  • @mugdha4128
    @mugdha41285 ай бұрын

    Man thank you so much for this video you are a legend i have a assessment on this topic and it is worth half my grade aand i didnt understand a thing my teacher taught me vut after watching this video i understand so much that i might just ace the assignment thx so much

  • @1stClassMaths

    @1stClassMaths

    5 ай бұрын

    Fantastic!

  • @new-qo8rb
    @new-qo8rb3 ай бұрын

    I get so confused -- how do i remember all this? i simply will forget 😔😔

  • @faizaax4376
    @faizaax43766 ай бұрын

    omg i went on like 50 websites trying to get help but this video saved mee

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    50, that’s a lot!

  • @tushiglutbekh5453
    @tushiglutbekh54535 ай бұрын

    you made the problem easy to solve😁😁

  • @bangchanstoez
    @bangchanstoez5 ай бұрын

    This helped so much thank you sooook much❤️❤️❤️❤️❤️

  • @1stClassMaths

    @1stClassMaths

    5 ай бұрын

    No problem 😊

  • @rizwanafzal7076
    @rizwanafzal70766 ай бұрын

    Very good.thanks for such good video. Very useful.

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    You are most welcome

  • @supariley
    @suparileyАй бұрын

    THANK YOU SO MUCH

  • @1stClassMaths

    @1stClassMaths

    Ай бұрын

    NO PROBLEM

  • @olusola9757
    @olusola97572 ай бұрын

    Thank you so much 😀

  • @hamzasherasmat6593
    @hamzasherasmat65936 ай бұрын

    sorry to ask, but how did you find a sequence for 3n^2 on the first one, do i just square root like the first one i have to square root, then the second one, then the third one?

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    Don't apologise! For the 3n^2 sequence you do the square numbers so 1,4,9,16,25 then multiply them all by 3.

  • @hamzasherasmat6593

    @hamzasherasmat6593

    6 ай бұрын

    @@1stClassMaths Ohhhh, thanks sir!

  • @usmanayoub1434
    @usmanayoub1434Ай бұрын

    Thank you it's working

  • @S21fe.hype0
    @S21fe.hype02 ай бұрын

    Thank you so much I now understand and tomorrow I have exam you deserve a sub

  • @1stClassMaths

    @1stClassMaths

    2 ай бұрын

    Thanks for the sub and good luck!

  • @Bollywoodwiddeepak
    @BollywoodwiddeepakАй бұрын

    I was trying to find the Nth term of quadratic equation from last two days but could not understand. But you explained it very well in short method 👍

  • @arqhur
    @arqhur2 ай бұрын

    thank you!

  • @user-cm7gf2yh3n
    @user-cm7gf2yh3nАй бұрын

    sometimes theres a sub question after you find the nth term of the sequence where it asks you to prove if a number is apart of the nth term of the sequence that you just found out. how do you do those sort of questions? (also thanks for explaining this topic so well it makes alot more sense now)

  • @zafarziya-ip3dq
    @zafarziya-ip3dq7 ай бұрын

    Thank you so much ❤

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    You're welcome 😊

  • @xmayx2507
    @xmayx25072 ай бұрын

    THANK YOU

  • @basselmohamed5680
    @basselmohamed56806 ай бұрын

    Your video is very helpful

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    Glad you think so!

  • @ratchet7748
    @ratchet77482 ай бұрын

    Thanks so much!

  • @1stClassMaths

    @1stClassMaths

    2 ай бұрын

    You're welcome!

  • @basselmohamed5680
    @basselmohamed56806 ай бұрын

    The last example is 2nsqure+-3n+5 not 2nsqure-3n+5 ,you forget to put the plus, if you want to work out the first term of the original sequence by using 2nsqure-3n+5 it will be 2-2 which is 0 but if use 2nsqure+-3n+5 it will be 2+2 which 4, in all cases we adding because the third sequence plus the second sequence equal the first sequence

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    Hi. Unfortunately you are not correct here. 2n^2 + - 3n + 5 is equivalent to 2n^2 - 3n + 5 (which is the answer). When you add a negative it is just the same as subtracting. Your errors comes in your substitution. If you substitute in n = 1 you get 2*(1)^2 - 3(1) + 5 = 2 - 3 + 5 = 4

  • @gracehathaway2715

    @gracehathaway2715

    6 ай бұрын

    Get told😂

  • @toneless4345

    @toneless4345

    Ай бұрын

    @@1stClassMaths why minus first, what about bedmas, 2-3+5, don't u have to add 3+5 first, which would give u 8?

  • @remarkablething5874
    @remarkablething58748 ай бұрын

    cheers mate

  • @user-og6vl2ms2s
    @user-og6vl2ms2s6 ай бұрын

    thank you very much bro.............

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    Thank you too :)

  • @Pegasus10707
    @Pegasus107076 ай бұрын

    thank you 😊

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    You're welcome 😊

  • @jacobhaile383
    @jacobhaile3833 ай бұрын

    On the second sequence where did you get the square numbers from

  • @1stClassMaths

    @1stClassMaths

    3 ай бұрын

    Square numbers are from 1x1, 2x2, 3x3 etc

  • @pen7822
    @pen78223 ай бұрын

    god bless you wonderful man

  • @1stClassMaths

    @1stClassMaths

    3 ай бұрын

    Thanks!

  • @hutt1232
    @hutt12323 ай бұрын

    Hello, why doesn’t the quadratic sequence formula work here? I do an^2 + bn + c but i get incorrect answers while in other questions i use the formula and its correct

  • @1stClassMaths

    @1stClassMaths

    3 ай бұрын

    Hi. I am not unsure what you mean? Can you give an example and I will try see what's happening :)

  • @shayanghasemi1993
    @shayanghasemi199324 күн бұрын

    Thanks a lot

  • @1stClassMaths

    @1stClassMaths

    21 күн бұрын

    No worries!

  • @samaaa8400
    @samaaa84006 ай бұрын

    THANK U SM

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    No problem

  • @MarieAddo-Tetteh
    @MarieAddo-Tetteh5 ай бұрын

    Thank you😊

  • @1stClassMaths

    @1stClassMaths

    5 ай бұрын

    You're welcome 😊

  • @disappointedshrek8681
    @disappointedshrek86812 ай бұрын

    (second difference/2)×n² square numbers multiplied by 2 (including 1) subtract the original sequence by the square numbers multiplied by 2 find the general rule for the new linear sequence brought forth by subtracting the two (first difference×n + first term-x=first difference)

  • @zeemish
    @zeemish3 ай бұрын

    thanks!

  • @1stClassMaths

    @1stClassMaths

    3 ай бұрын

    No problem!

  • @seblemekete8413
    @seblemekete84133 ай бұрын

    Thank you so mach😅 you save ME if i donot see this vidio i will get

  • @shahdahmed6436
    @shahdahmed64363 ай бұрын

    THANKYOUTHANKYOUTHANKYOUTHANKYOU!!!

  • @1stClassMaths

    @1stClassMaths

    3 ай бұрын

    NP

  • @andrewr3243
    @andrewr324328 күн бұрын

    What happens if 2nd difference is an odd number like 5? Would nth term be written starting with 2.5n

  • @1stClassMaths

    @1stClassMaths

    28 күн бұрын

    Yes

  • @andrewr3243

    @andrewr3243

    28 күн бұрын

    @1stClassMaths wow thanks for the super swift reply!! 😁😁

  • @deonxholi9800
    @deonxholi98005 ай бұрын

    why is it 4n -1 but not for ex 4n+1

  • @nicolaeveritt5555
    @nicolaeveritt55558 ай бұрын

    Thanks so much

  • @1stClassMaths

    @1stClassMaths

    8 ай бұрын

    You're welcome!

  • @wikaarad
    @wikaarad4 ай бұрын

    This was genuinely so helpful and clear thank you 🙏🏼

  • @1stClassMaths

    @1stClassMaths

    4 ай бұрын

    You're welcome!

  • @wewiee
    @wewiee4 ай бұрын

    What if the second difference is not constant?

  • @1stClassMaths

    @1stClassMaths

    4 ай бұрын

    The sequence is not a quadratic sequence.

  • @user-so5qr5jh8p
    @user-so5qr5jh8p6 ай бұрын

    What if the answer after subtracting the sequence is a sequence that has the same number

  • @1stClassMaths

    @1stClassMaths

    6 ай бұрын

    Great question. This just means the nth term of it is simply the constant number. So if it's always 3, 3, 3, 3, your nth term might be something like n^2 + 3

  • @user-so5qr5jh8p

    @user-so5qr5jh8p

    6 ай бұрын

    @@1stClassMaths ohhh thank you so much❤

  • @NpcFR01
    @NpcFR018 ай бұрын

    What about having 3 layer on the sequence?

  • @1stClassMaths

    @1stClassMaths

    8 ай бұрын

    It would be a cubic sequence

  • @NpcFR01

    @NpcFR01

    8 ай бұрын

    @@1stClassMaths is that any method to solve it?Btw thank you for your response

  • @1stClassMaths

    @1stClassMaths

    8 ай бұрын

    Hi yes there is but it is not needed for GCSE. Search for cubic nth term videos. 😀

  • @RoBotOnd
    @RoBotOnd8 ай бұрын

    5:16 how did u get the +5!? Pls 😢answer brother i got an exam tomorrow

  • @1stClassMaths

    @1stClassMaths

    8 ай бұрын

    To get from -3 to +2 (the first term) we add 5

  • @RoBotOnd

    @RoBotOnd

    8 ай бұрын

    @@1stClassMaths Thank you so much!! ❤️

  • @educationservices7426
    @educationservices74265 ай бұрын

    Why we use the 1,4,9,16,25 values to multiply with woth to get the answer 3,12,27,48,75...kindly anyone plz help me at this stage bcz I didn't get the point from where we get 1,4,9,16,25 values

  • @1stClassMaths

    @1stClassMaths

    5 ай бұрын

    They are the square numbers.

  • @educationservices7426

    @educationservices7426

    5 ай бұрын

    @@1stClassMaths ...and in the last how he get -1 that is the value of c he find out ..how it comes?

  • @s4-yt
    @s4-yt2 ай бұрын

    10 mins before my exam 😂

  • @omarbouhassoun1775
    @omarbouhassoun17752 ай бұрын

    In the second example it’s supposed to be 3n-1 not 3n+5

  • @1stClassMaths

    @1stClassMaths

    2 ай бұрын

    Well it is actually neither of those. It is -3n + 5 since -3 + 5 = 2

  • @Yche-21M
    @Yche-21M7 ай бұрын

    bro how you did that 2:18 3n²

  • @1stClassMaths

    @1stClassMaths

    7 ай бұрын

    Bro not sure what you mean? Where did the 3n^2 come from or how did I get the numbers 3, 12, 27, 48 and 75?

  • @nicole27777

    @nicole27777

    7 ай бұрын

    because you halve the second difference (6)

  • @KOZBON-lv8zy
    @KOZBON-lv8zy2 ай бұрын

    i did this question and I got 2nsquared - 1n + 3, can you tell me what i did wrong? - (2,7,16,29,46)

  • @1stClassMaths

    @1stClassMaths

    2 ай бұрын

    It is 2n^2 - n + 1

  • @KOZBON-lv8zy

    @KOZBON-lv8zy

    2 ай бұрын

    @@1stClassMaths Oh really? Chat gpt is wrong then 🤣 THANK YOU!!!

  • @1stClassMaths

    @1stClassMaths

    2 ай бұрын

    chatgpt is pretty bad at maths.

  • @frigid.editzz
    @frigid.editzz2 ай бұрын

    I DID understand

  • @charcoalotte2021
    @charcoalotte20214 ай бұрын

    Okay so the thing is my sequence is 4,9,16,25 how do i do that?

  • @1stClassMaths

    @1stClassMaths

    4 ай бұрын

    Same way. The second difference is 2 so you have n^2. Subtract this (1,4,9,16,25) from the sequence and you get 3,5,7,9,11 The nth term of the linear part is 2n+1 so n^2 + 2n + 1

  • @alescuzyehhh

    @alescuzyehhh

    2 ай бұрын

    @@1stClassMathsoh wow you replied to soo many comments!! that mustve taken a long time ahah.. thank you so much for this video, its so helpful.. i was struggling to understand it in my class but this made it so much easier for me!!

  • @1stClassMaths

    @1stClassMaths

    2 ай бұрын

    I try to answer as many of the questions as I can.

  • @alescuzyehhh

    @alescuzyehhh

    2 ай бұрын

    @@1stClassMaths 😂😂😂

  • @sephorarufai
    @sephorarufai4 ай бұрын

    thank you broo😭 my teachers can’t explain it properly. but why tf would i need to know how to find an nth term?? it’s so long asw like what the heck

  • @k.kai2434

    @k.kai2434

    4 ай бұрын

    Ikr idek why but at least i can do it now💀

  • @StrikingStew
    @StrikingStewАй бұрын

    saved the paper 2

  • @falingunit
    @falingunit5 ай бұрын

    you can actually apply the same idea to find out the nth term of any polynomial

  • @LovelearningAgain
    @LovelearningAgain2 ай бұрын

    Thanks a bunch. To the point. If you don't mind, what is the name of the software that you used in this video?

  • @1stClassMaths

    @1stClassMaths

    2 ай бұрын

    Thanks. I use powerpoint.

  • @wandaxmayfield14
    @wandaxmayfield14Ай бұрын

    I dont grt why u added 5 on the 2nd equation

  • @jamespiggin4612
    @jamespiggin4612Ай бұрын

    Class video

  • @1stClassMaths

    @1stClassMaths

    Ай бұрын

    1st Class

  • @Sadisfynotfound
    @Sadisfynotfound2 ай бұрын

    5:06 i dont understand the linear part because 2(1)^2 is 4 but you said 2!!!!!!!

  • @Sadisfynotfound

    @Sadisfynotfound

    2 ай бұрын

    ok i understand now thx

  • @talharafaqat3084
    @talharafaqat3084Ай бұрын

    I didnt get a linear sequence the question was find the nth term of the sequence 4,9,16,25,36

  • @user-ek7bq8tp7v
    @user-ek7bq8tp7v5 ай бұрын

    I didn't understand

  • @galaxyz3709

    @galaxyz3709

    2 ай бұрын

    Practice and you will get it

  • @alescuzyehhh

    @alescuzyehhh

    2 ай бұрын

    @@galaxyz3709yep!!

  • @waleedmalik785

    @waleedmalik785

    2 ай бұрын

    If you want a easier way to find b,the equation 3a + b=the first term of the first sequence.It will give you value of b,you can find the value of c using the quadratic equation,an2+bn+c,put the value of a and b and find c,btw the 2 in an2 represents the n being squared.

  • @pces.x

    @pces.x

    Ай бұрын

    I don’t, I don’t get it

  • @GamingWithJas-ff4yf

    @GamingWithJas-ff4yf

    Ай бұрын

    What its so easy

  • @user-rubbishsoup
    @user-rubbishsoup6 ай бұрын

    even with the best explaination vid about nth term Quadratic Sequence, it still sounds complicated enough. This shows how stupid this tpoic is

  • @k.kai2434

    @k.kai2434

    5 ай бұрын

    Yeah 💀

  • @drrkwolf
    @drrkwolf2 ай бұрын

    a+b+c = first original term 3a + b = 1st diff first term 2a = 2nd diff first term a+b+c = 8, 12, 20, 32, 48 3a+b = 4, 8, 12, 16 2a = 4, 4, 4 a = 2, b = -2 c = 8 2n^2 - 2n + 8

  • @L0St_Memories0

    @L0St_Memories0

    2 ай бұрын

    yea i was searching for this

  • @omnath3577
    @omnath35774 ай бұрын

    Anybody can solve this Find the nth term of the sequence 3,5,10,22,47,93

  • @abdelrahmanmohamad9190
    @abdelrahmanmohamad91904 ай бұрын

    I suck at this chapter

  • @k.kai2434

    @k.kai2434

    4 ай бұрын

    Same💀

  • @Twilamation4643
    @Twilamation46433 ай бұрын

    Not really helping but at least I have an idea on what the nth term of 5 7 11 14 19 is

  • @xlldx
    @xlldxАй бұрын

    is it just me or does this not help AT ALL😭🙏🏾

  • @1stClassMaths

    @1stClassMaths

    Ай бұрын

    Maybe just you :(

  • @HassanKhan-oc6fp
    @HassanKhan-oc6fpАй бұрын

    Thanks!

  • @trabbelsiskyee
    @trabbelsiskyee Жыл бұрын

    THANK YOU SO MUCH

  • @1stClassMaths

    @1stClassMaths

    Жыл бұрын

    You're welcome!

  • @user-dm4sx6kt8k
    @user-dm4sx6kt8k10 ай бұрын

    Thanks!

  • @1stClassMaths

    @1stClassMaths

    8 ай бұрын

    No problem!

Келесі