Showing a sequence is less than another sequence (sequences and series question)

Therefore, an+1 is always less than an.In summary, we can show that whatever value of an+1 we substitute, it will always be less than an. This is because when rearranging the original equation and factoring it, we get an inequality that proves an+1 is always less than an. Therefore, we can conclude that the sequence is less than another sequence.
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!Showing a sequence is less than another sequence (sequences and series question)

Homework Statement



[PLAIN]http://img263.imageshack.us/img263/385/sandq1.gif

Homework Equations


The Attempt at a Solution



i re arranged the equation and wrote in terms of an

to get an=-4an+1/an+1-7

and an+1 can not be 7

but i don't know how to show that, whatever value of an+1 i sub, it will always be less than an
 
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is this proof?

Let an = 3 + x where x is positive.

an+1= 7(3 + x) / (3 + x + 4) = (21 + 7x)/(7 + x)

= (21 + 3x + 4x) / (7 + x) = 3 + 4x/(7 + x) > 3

re arrange and factorise ==> an>0 and an>3

This is because the last part 4x/(7 + x) must be positive if x is positive.

Now suppose an+1 = 7*an / (an + 4) <= an. What would follow?an > 3 ----> (an)2 > 3*an ----> (an)^2 + 4*an > 7*an

Divide by (an + 4) to get an > 7an / (an + 4)

an > an+1 or an+1 < an
 
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FAQ: Showing a sequence is less than another sequence (sequences and series question)

What is a sequence?

A sequence is a list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term, and the position of a term in the sequence is called its index.

How do you show that one sequence is less than another?

To show that one sequence is less than another, you can use the property of monotonicity. This means that if every term in the first sequence is less than or equal to the corresponding term in the second sequence, then the first sequence is less than the second sequence.

What is the difference between a sequence and a series?

A sequence is a list of numbers, while a series is the sum of the terms in a sequence. For example, the sequence 1, 2, 3, 4, 5 would have a series of 1+2+3+4+5=15.

Can you compare sequences with different numbers of terms?

Yes, you can still compare sequences with different numbers of terms. You can extend the shorter sequence by repeating its last term until it has the same number of terms as the longer sequence. Then, you can compare the extended sequences using the property of monotonicity.

What are some common patterns or rules in sequences?

Some common patterns or rules in sequences include arithmetic sequences, where the difference between consecutive terms is constant, and geometric sequences, where the ratio between consecutive terms is constant. Other patterns include Fibonacci sequences, which add the previous two terms to get the next term, and factorial sequences, where each term is the product of all previous terms.

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