# maths iteration homework

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Consider sequences a1, a2, a3, . . . of positive real numbers with a1 = 1 and such that an+1 + an = (an+1 − an)^2 for each positive integer n. How many possible values can a2017 take?

I am really confused on this question, I have tried to find out what a2 is, but I got (a2)^2-2a2. I then found out what was being added to each term of the sequence ( I got (a2)^2-2a2-1 ) Finally i substituted it into the sequence to get a2017 = 2016(a2)^2 - 4032a2 - 2015. However, because I can't find a value for a2, I can't find out what a2017 is. If someone could please tell me what I have done wrong or how I can find other values, it would be great.

Thank you.

I am really confused on this question, I have tried to find out what a2 is, but I got (a2)^2-2a2. I then found out what was being added to each term of the sequence ( I got (a2)^2-2a2-1 ) Finally i substituted it into the sequence to get a2017 = 2016(a2)^2 - 4032a2 - 2015. However, because I can't find a value for a2, I can't find out what a2017 is. If someone could please tell me what I have done wrong or how I can find other values, it would be great.

Thank you.

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#2

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Consider sequences a1, a2, a3, . . . of positive real numbers with a1 = 1 and such that an+1 + an = (an+1 − an)^2 for each positive integer n. How many possible values can a2017 take?

I am really confused on this question, I have tried to find out what a2 is, but I got (a2)^2-2a2. I then found out what was being added to each term of the sequence ( I got (a2)^2-2a2-1 ) Finally i substituted it into the sequence to get a2017 = 2016(a2)^2 - 4032a2 - 2015. However, because I can't find a value for a2, I can't find out what a2017 is. If someone could please tell me what I have done wrong or how I can find other values, it would be great.

Thank you.

**Hannahd2003**)Consider sequences a1, a2, a3, . . . of positive real numbers with a1 = 1 and such that an+1 + an = (an+1 − an)^2 for each positive integer n. How many possible values can a2017 take?

I am really confused on this question, I have tried to find out what a2 is, but I got (a2)^2-2a2. I then found out what was being added to each term of the sequence ( I got (a2)^2-2a2-1 ) Finally i substituted it into the sequence to get a2017 = 2016(a2)^2 - 4032a2 - 2015. However, because I can't find a value for a2, I can't find out what a2017 is. If someone could please tell me what I have done wrong or how I can find other values, it would be great.

Thank you.

_{n+1}(treating a

_{n}as a constant), and solving using the formula or completing the square. The easier way would be to put a

_{1}= 1 in the equation and solve for a

_{2}. This would give you a

_{2}+ 1 = (a

_{2}- 1)

^{2}, which is pretty straightforward to solve.

Despite the unusual way that this is given in, it has been set up so that the sequence is quite well behaved.

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#3

**Hannahd2003**)

Consider sequences a1, a2, a3, . . . of positive real numbers with a1 = 1 and such that an+1 + an = (an+1 − an)^2 for each positive integer n. How many possible values can a2017 take?

I am really confused on this question, I have tried to find out what a2 is, but I got (a2)^2-2a2. I then found out what was being added to each term of the sequence ( I got (a2)^2-2a2-1 ) Finally i substituted it into the sequence to get a2017 = 2016(a2)^2 - 4032a2 - 2015. However, because I can't find a value for a2, I can't find out what a2017 is. If someone could please tell me what I have done wrong or how I can find other values, it would be great.

Thank you.

which, if you solve, yields

You're told that .

So, hence .

What about ?

Can you spot the pattern ?

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You can expand into the form

which, if you solve, yields

You're told that .

So, hence .

What about ?

Can you spot the pattern ?

**RDKGames**)You can expand into the form

which, if you solve, yields

You're told that .

So, hence .

What about ?

Can you spot the pattern ?

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#5

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So, if a1 has 1 value, a2 has 2 values, a3 has 4 values... So the number of values equals 2n-2, so if n=2017, the number of values is 4032?

**Hannahd2003**)So, if a1 has 1 value, a2 has 2 values, a3 has 4 values... So the number of values equals 2n-2, so if n=2017, the number of values is 4032?

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Did you calculate a3 values explicitly, or are you guessing that it has 4?

**RDKGames**)Did you calculate a3 values explicitly, or are you guessing that it has 4?

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#7

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I just guessed, but I think that I may need to work it out and draw some sort of diagram?

**Hannahd2003**)I just guessed, but I think that I may need to work it out and draw some sort of diagram?

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So, if a1 = 1 and a2 = 0 or 3, is it possible for a3 to equal 1 or 0?

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#9

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So, if a1 = 1 and a2 = 0 or 3, is it possible for a3 to equal 1 or 0?

**Hannahd2003**)So, if a1 = 1 and a2 = 0 or 3, is it possible for a3 to equal 1 or 0?

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Why wouldn't it be?

**RDKGames**)Why wouldn't it be?

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#11

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Because an arithmetic sequence goes up/down by the same number each time, so if the sequence goes 1,0... It goes down by one each time so the third term couldn't be 0. Just a thought.

**Hannahd2003**)Because an arithmetic sequence goes up/down by the same number each time, so if the sequence goes 1,0... It goes down by one each time so the third term couldn't be 0. Just a thought.

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#12

**Hannahd2003**)

So, if a1 = 1 and a2 = 0 or 3, is it possible for a3 to equal 1 or 0?

__positive__real numbers, so 0 is not acceptable. Also, if 1 and 0 are your only choices for a3, then you're gone wrong somewhere.

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(Original post by

Perhaps worth commenting that the sequence is of

**ghostwalker**)Perhaps worth commenting that the sequence is of

__positive__real numbers, so 0 is not acceptable. Also, if 1 and 0 are your only choices for a3, then you're gone wrong somewhere.
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So I have now realised that all of the numbers are triangular, so I have used gauss' formula to find out the 2017th triangular number (which was 2035153) and then I divided it by 2017 to get the number of values as 1009.

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#15

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So I have now realised that all of the numbers are triangular, so I have used gauss' formula to find out the 2017th triangular number (which was 2035153) and then I divided it by 2017 to get the number of values as 1009.

**Hannahd2003**)So I have now realised that all of the numbers are triangular, so I have used gauss' formula to find out the 2017th triangular number (which was 2035153) and then I divided it by 2017 to get the number of values as 1009.

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#16

**Hannahd2003**)

So I have now realised that all of the numbers are triangular, so I have used gauss' formula to find out the 2017th triangular number (which was 2035153) and then I divided it by 2017 to get the number of values as 1009.

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(Original post by

Just to point out now you've got the answer...you didn't need to work out that large triangular number - since the formula is (n/2)(n + 1) and you're dividing by n again (in this case 2017) you just needed to work out (1/2) (2018) = 1009

**davros**)Just to point out now you've got the answer...you didn't need to work out that large triangular number - since the formula is (n/2)(n + 1) and you're dividing by n again (in this case 2017) you just needed to work out (1/2) (2018) = 1009

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