MHB Graeme's YAnswers Question: The effect of changing values in sequences?

[CaptainBlack]

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?

.

 

[CaptainBlack]

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]

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