We do not need all 3 formulas. In fact, using the infinite sum formula is enough. Notice that S∞ - Sn is just the sum from the (n+1)-th term to ∞-th term, so: S∞ - Sn = t(n + 1) + t(n + 2) + ... = t(n + 1)/(1 - r), for |r| < 1 = tn × r/(1 - r) Therefore, tn/(S∞ - Sn) = (1 - r)/r = 2/3 Solving gives r = 3/5 < 1.
@@almightyhydraIt's like on physics where each moment of a movement could be regarded as the start point of the movement (so all parameters could be transformed to initial ones - velocity to initial velocity, time and distance to 0).
This is what I did: -> rearranging , Sn =Sinfin -3/2tn S1= t1 [since that's just the 1st term] Also, Sinfin= t1/1-r S1=Sinfin-3/2t1 S1= t1/1-r -3/2t1 t1 = (t1/1-r) -3/2t1,now we have eqn in terms of t1 and r t1+3/2t1 = t1/1-r 5/2 = 1/1-r R is 3/5
I know somewhere out there is someone who after watching this video and not understanding a part (or perhaps most of it) is thinking that he/she isn't meant for maths but that's not true as long as you are consistent with giving time to the subject you will be able to understand it, sure right now you may not be able to keep up with the pace at your school/uni but if you begin learning at your own pace after certain time you will be able to go at faster pace than your peers , so start down with basics, doesn't matter if it's precalc or pre algebra you are bad at, onc you get the basics done you will be able to understand the stuff you once thought wasn't meant for you Tip : pick up a specific book for learning that topic, they are simply best source you can refer to which often includes proofs and reason why things are the way they are which is often skipped in classrooms (unless your instructor is really good ( bless those rare entities 🛐)) ( I wish someone had told me this like 6-7 years ago xd)
2:45 I would love to see a proof for this! A lot of the time in high school math anyway, teachers skip over how formulas are created and it really annoys me because I feel knowing how the formula works gives me a better understanding of it. So if you can please make this video in the future. Thanks!
This one was sooo fun to do and easy to as long as you're not "scared" when you first see the question love it! Second i got scared in the starting although when i arranged formulas together and did it pretty well it all made sense, last thing one must know how exponentials work!
6:45 can't you just say since it's true for all N just pick N=1 and plug and chug? edit: Yep, 1-r/r=2/3 is 3/5 In fact, couldn't you have done this at 4:20? edit: yep, you can do it at 4:20, too (WA for "t/((t/(1-r))-(t(1-r)/(1-r))) = 2/3 solve for r" = 3/5 I wish it didn't delete links to wolfram alpha from comments
Open Excel Assign a cell for the constant ratio, which we'll call r_. Note that r itself is reserved, and can't be used. Assign the name r_ to it. Enter in a number less than 1, which we'll modify later, for the ratio r. Create a calculated constant in another cell, assign the name Sinf, which will equal 1/(1 - r_) Start a column equal to 1 as our initial sequence value. Let's suppose you start in cell A10 In the next row down, cell A11, specify the formula =A10*r_ Pull down this formula, to fill several more rows (perhaps 10 rows) In cell B10, type =sum(A$10:A10), and pull it down as much as you can. this produces the partial sum, Sn. Observe that the bigger the sum gets, the closer it gets to converging to Sinf. In Cell C10, type =Sinf - B10. This calculates the infinite series result Sinf, minus the partial sum Sn. In Cell D10, type =A10/C10, and pull it down Observe that all your D-column values are the same. Adjust the value of r_, until the D-column values match what we were given in this problem.
The things we should already realise about this problem are that: 1: it's geometric. Meaning every next number is multiplied by a value r 2: it converges, so that r is less than 1, otherwise it would not be possible to infinitely add the terms together for a sum. From here, it's really just about knowing what the formulas for adding the terms in the sequence is. If you understand how to get each individual component of the above formula, then it becomes a simple algebra problem.
Proofs of these formulas: 👉 ua-cam.com/video/DwJ1LH9gygk/v-deo.html
We do not need all 3 formulas. In fact, using the infinite sum formula is enough.
Notice that S∞ - Sn is just the sum from the (n+1)-th term to ∞-th term, so:
S∞ - Sn
= t(n + 1) + t(n + 2) + ...
= t(n + 1)/(1 - r), for |r| < 1
= tn × r/(1 - r)
Therefore,
tn/(S∞ - Sn) = (1 - r)/r = 2/3
Solving gives r = 3/5 < 1.
good spot - a subsequence of a geometric series is itself a geometric series
Wow😮
@@almightyhydraIt's like on physics where each moment of a movement could be regarded as the start point of the movement (so all parameters could be transformed to initial ones - velocity to initial velocity, time and distance to 0).
This is what I did:
-> rearranging , Sn =Sinfin -3/2tn
S1= t1 [since that's just the 1st term]
Also, Sinfin= t1/1-r
S1=Sinfin-3/2t1
S1= t1/1-r -3/2t1
t1 = (t1/1-r) -3/2t1,now we have eqn in terms of t1 and r
t1+3/2t1 = t1/1-r
5/2 = 1/1-r
R is 3/5
I know somewhere out there is someone who after watching this video and not understanding a part (or perhaps most of it) is thinking that he/she isn't meant for maths but that's not true as long as you are consistent with giving time to the subject you will be able to understand it, sure right now you may not be able to keep up with the pace at your school/uni but if you begin learning at your own pace after certain time you will be able to go at faster pace than your peers , so start down with basics, doesn't matter if it's precalc or pre algebra you are bad at, onc you get the basics done you will be able to understand the stuff you once thought wasn't meant for you
Tip : pick up a specific book for learning that topic, they are simply best source you can refer to which often includes proofs and reason why things are the way they are which is often skipped in classrooms (unless your instructor is really good ( bless those rare entities 🛐))
( I wish someone had told me this like 6-7 years ago xd)
You are genius calculus
2:45 I would love to see a proof for this! A lot of the time in high school math anyway, teachers skip over how formulas are created and it really annoys me because I feel knowing how the formula works gives me a better understanding of it. So if you can please make this video in the future. Thanks!
Very nice explanation
This one was sooo fun to do and easy to as long as you're not "scared" when you first see the question love it! Second i got scared in the starting although when i arranged formulas together and did it pretty well it all made sense, last thing one must know how exponentials work!
6:45 can't you just say since it's true for all N just pick N=1 and plug and chug?
edit: Yep, 1-r/r=2/3 is 3/5
In fact, couldn't you have done this at 4:20? edit: yep, you can do it at 4:20, too (WA for "t/((t/(1-r))-(t(1-r)/(1-r))) = 2/3 solve for r" = 3/5
I wish it didn't delete links to wolfram alpha from comments
I did the exact same thing lol
JEE Aspirants welcome to this video❤❤
How do we get an empirical sense of this?
Open Excel
Assign a cell for the constant ratio, which we'll call r_. Note that r itself is reserved, and can't be used. Assign the name r_ to it. Enter in a number less than 1, which we'll modify later, for the ratio r.
Create a calculated constant in another cell, assign the name Sinf, which will equal 1/(1 - r_)
Start a column equal to 1 as our initial sequence value. Let's suppose you start in cell A10
In the next row down, cell A11, specify the formula =A10*r_
Pull down this formula, to fill several more rows (perhaps 10 rows)
In cell B10, type =sum(A$10:A10), and pull it down as much as you can. this produces the partial sum, Sn. Observe that the bigger the sum gets, the closer it gets to converging to Sinf.
In Cell C10, type =Sinf - B10. This calculates the infinite series result Sinf, minus the partial sum Sn.
In Cell D10, type =A10/C10, and pull it down
Observe that all your D-column values are the same. Adjust the value of r_, until the D-column values match what we were given in this problem.
S(inf)-S(n)
= ( t(1)/(1-r) ) - ( (t(1)(1-r^n) ) / (1-r) )
= ( (t(1)(1 - (1-r^n) ) ) / (1-r)
= ( t(1)r^n ) / (1-r)
= t(n+1) / (1-r)
= t(n) (r/(1-r)
So basically the whole thing is equal to ( (1-r) / r )
Oh wait, empirical sense. Sorry about that ignore me please.
The things we should already realise about this problem are that:
1: it's geometric. Meaning every next number is multiplied by a value r
2: it converges, so that r is less than 1, otherwise it would not be possible to infinitely add the terms together for a sum.
From here, it's really just about knowing what the formulas for adding the terms in the sequence is. If you understand how to get each individual component of the above formula, then it becomes a simple algebra problem.
@@carultch I solved the problem on my own but couldn't understand what you wrote
I guess I would stick to the pen and paper
Your r looks like a Greek alphabet gamma.
It's messing me up.
Just put n =1
When we put n =1 we get s of n = t of n
Now just use one formula that is s of infinity ♾️ you will get the same answer that is ⅗
What he did makes much more sense than randomly assigning a value to n
But for any n it is true 👍and on top of that n =1 is the most suitable value if we want to calculate the common ratio
@@kevv.1912actually assigning a value make sense if your answer doesn't depends on the value to n