Graeme's Y!Answers Question: The effect of changing values in sequences??

CaptainBlack

Well-known member
The effect of changing values in sequences??

I have been given a maths assignment and have been given equations $$u_{n+1}=2u_{n}+2$$ and asked what is the effect if the value $$u_{0}$$ is changed? I used multiple values both positive and negative and have only noticed taht when it is a high negative number it flips up side down?

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CaptainBlack

Well-known member
Look at what happens as you expand the sequence:

$$u_{1} = 2 u_{0}+2$$
$$u_{2} = 2( 2 u_{0}+2 ) + 2 =2^2 u_{0} + 2^2 + 2$$
$$u_{3} = 2( 2^2 u_{0} + 2^2 + 2 ) + 2 = 2^3 u_{0} + 2^3 + 2^2 + 2$$
:
:
$$u_{n} = 2^n u_{0} + 2^n + 2^{n-1} + ... + 2^2 + 2$$

The earlier terms in the sequence above suggest the last one, which can be easily proven by induction.

The last term can now be simplified to:

$$u_{n} = 2^n u_{0} + 2^{n+1} - 2$$

So you now have all the information you need to answer questions about how the sequence depends on $$u_{0}$$.

QuestForInsight

Member
An alternative would be generating functions (which I'm quite fond of).