Calculating Terms of Sequence: a1,a2,a3,a4

In summary, the given sequence has a closed form of A_n=550 and the values of a1, a2, a3, and a4 are all equal to 550. There is only one stable point at x=550, making the sequence easy to calculate.
  • #1
Logan Land
84
0
write the terms a1,a2,a3,a4 of the following sequence. an+1=0.4an+330, a0=550

everytime I get 550 for a1 a2 a3 and a4
is that correct or am I doing it wrong.
 
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  • #2
550 * 0.4 = 220

220 + 330 = 550

Pretty clear. Why do you doubt?
 
  • #3
It just seemed to easy for Calculus that's all.

Thanks
 
  • #4
If we write the inhomogeneous recursions:

\(\displaystyle a_{n+1}=0.4a_{n}+330\)

\(\displaystyle a_{n+2}=0.4a_{n+1}+330\)

We find the homogenous recursion via differencing:

\(\displaystyle a_{n+2}=1.4a_{n+1}-0.4a_{n}\)

whose associated characteristic roots are:

\(\displaystyle r=\frac{2}{5},\,1\)

and hence the closed form is:

\(\displaystyle A_n=k_1+k_2\left(\frac{2}{5} \right)^n\)

Now, using:

\(\displaystyle A_0=A_1=550\), we may determine:

\(\displaystyle k_1=550,\,k_2=0\) and so:

\(\displaystyle A_n=550\)
 
  • #5
LLand314 said:
write the terms a1,a2,a3,a4 of the following sequence. an+1=0.4an+330, a0=550

everytime I get 550 for a1 a2 a3 and a4
is that correct or am I doing it wrong.

Using the procedure described in...

http://www.mathhelpboards.com/f15/difference-equation-tutorial-draft-part-i-426/

... the difference equation can be written as...

$\displaystyle \Delta_{n}= a_{n+1}-a_{n} = 330 - .6\ a_{n} = f(a_{n})$ (1)

... and the function f(*) is represented here...

http://www.123homepage.it/u/i68681865._szw380h285_.jpg.jfif
There is only one attractive fixed point in x=550 and, because the linearity of f(*) the stable point is met at the first step...

Kind regards

$\chi$ $\sigma$
 

1. How do you calculate the nth term of a sequence?

To calculate the nth term of a sequence, you can use the formula an = a1 + (n-1)d, where a1 is the first term, d is the common difference, and n is the term number you want to find.

2. What is a common difference in a sequence?

The common difference in a sequence is the constant value that is added or subtracted from one term to the next. It helps determine the pattern and allows you to find any term in the sequence.

3. How do you determine if a sequence is arithmetic or geometric?

To determine if a sequence is arithmetic or geometric, you can look at the differences between each term. In an arithmetic sequence, the differences will be constant, while in a geometric sequence, the ratios between each term will be constant.

4. Can a sequence have more than one pattern?

Yes, a sequence can have more than one pattern. This is known as a mixed sequence. In this case, you would need to identify the different patterns and use them to find the missing terms.

5. How can sequences be used in real-life situations?

Sequences can be used in various real-life situations such as predicting stock market trends, calculating population growth, and determining the depreciation of assets. They can also be used in computer algorithms and coding.

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