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By "the nth term" of a sequence we mean an expression that will allow us to calculate the term that is in the nth position of the sequence. For example consider the sequence 2, 4, 6, 8, 10,... The pattern is easy to see. # The first term is two. # The second term is two times two. # The third term is two times three. # The fourth term is two times four. # The tenth term is two times ten. # the nineteenth term is two times nineteen. # The nth term is two times n.

In this sequence the nth term is 2n.

Q: Prove that no matter what the real numbers a and b are the sequence with nth term a nb is always an AP?

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No. No matter how large of an example you choose, someone always can find a larger number (of any kind), because the upper range of number is infinite. If you take all the known prime numbers and multiply them together, then add 1 to the result, you will have a number that is not divisible by any of the known prime numbers. This number will either be prime or have prime factors that were not previously known. So, in this way, you can always find a new prime number or a number that is a multiple of new prime numbers. If the known prime numbers include all the prime numbers up to the largest known, the new ones must be larger.

We prove that if an increasing sequence {an} is bounded above, then it is convergent and the limit is the sup {an }Now we use the least upper bound property of real numbers to say that sup {an } exists and we call it something, say S. We can say this because sup {an } is not empty and by our assumption is it bounded above so it has a LUB.Now for all natural numbers N we look at aN such that for all E, or epsilon greater than 0, we have aN > S-epsilon. This must be true, because if it were not the that number would be an upper bound which contradicts that S is the least upper bound.Now since {an} is increasing for all n greater than N we have |S-an|

They are not. They are countably infinite. That is, there is a one-to-one mapping between the set of rational numbers and the set of counting numbers.

Neither trigonometry nor any other subject can be used to prove questions. It may be possible to answer questions but that is another matter,

To prove a statement false, you need ONE example of when it is not true.To prove it true, you need to show it is ALWAYS true.

Related questions

If the number can be expressed as a ratio of two integer (the second not zero) then the number is rational. However, it is not always a simple matter to prove that if you cannot find such a representation, then the number is not rational: it is possible that you have not looked hard enough!

if it has mass

You can use the comparison test. Since the convergent sequence divided by n is less that the convergent sequence, it must converge.

A logical sequence in an argument is a way to prove a step has a logical consequence. Every proposition in an argument must be tested in this fashion to prove that every action has a reaction.

Real small

Inflate a balloon.

A hypothesis can never be proven true because there is always a possibility that it can be disproved. No matter how many times something happens, and how sure you think you are that it will happen again, there are always other possibilities that have not been explored.

No. No matter how large of an example you choose, someone always can find a larger number (of any kind), because the upper range of number is infinite. If you take all the known prime numbers and multiply them together, then add 1 to the result, you will have a number that is not divisible by any of the known prime numbers. This number will either be prime or have prime factors that were not previously known. So, in this way, you can always find a new prime number or a number that is a multiple of new prime numbers. If the known prime numbers include all the prime numbers up to the largest known, the new ones must be larger.

if they are always there if i need them.,

Any number you like. Since there is no sequence defined you can chose whatever you like and make the 7th term whatever you like. On a more serious note, though, given a set of six of fewer numbers, there are infinitely many different rules that will generate those six points and, by choosing the right one, you can make the seventh term whatever you like. The simplest way is to fit a polynomial of order 6 to the seven point (easy to prove this is always possible).

Why prove anything to people, who do not matter in your life, unless ofcourese they start suffocating you.

It has not yet been proven whether any arbitrary sequence of digits appears somewhere in the decimal expansion of pi.