Guessing Explicit Formula for a Sequence

In summary: Remember to always check your work and make sure it satisfies the initial conditions. In summary, the explicit formula for the given recursive sequence is a(n) = 5^(n-1)*(n-1) + 1.
  • #1
imdaman
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Homework Statement



Given a recursive sequence, use iteration to guess an explicit formula
a(k) = 5a(k-1) + k and a(1) = 1

Homework Equations



Sum of Geometric Sequence : (r^(n+1)-1)/(r-1)

The Attempt at a Solution



y(2) = 5*1 + 2

y(3) = 5(5+2) + 3 = 5^2 5*2 + 3

y(4) = 5(5^2 5*2 + 3) + 4 = 5^3 + 5^2*2 + 5*3 + 4

y(5) = 5(5^3 + 5^2*2 + 5*3 + 4) + 5 = 5^4 + 5^3*2 + 5^2*3 + 5*4 + 5

I see the pattern for the power is k-1 and decreases by 1 but the extra multiplication that increases by 1 is throwing me off.

Should I factor out a 5? I don't really see a pattern though.

I'm not sure if I'm going anywhere with this :
y(5) = 5^(n-1)*(n-4) + 5^(n-2)*(n-3) + 5^(n-3)*(n-2) + 5^(n-4)*(n-1) = 5^(n-1)*(n-(n-1)) + 5^(n-2)*(n-(n-2)) + 5^(n-3)*(n-(n-3)) + 5^(n-4)*(n-(n-4))

Any suggestions?
 
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  • #2


Hi there,

Thank you for sharing your attempt at solving this problem. It seems like you are on the right track with identifying the pattern for the power of 5. However, you are correct that the extra multiplication term is throwing off the pattern. In order to solve this, you can try to factor out a 5 from each term, as you mentioned.

Let's take a look at the first few terms:

a(1) = 1
a(2) = 5a(1) + 2 = 5*1 + 2 = 7
a(3) = 5a(2) + 3 = 5*7 + 3 = 38
a(4) = 5a(3) + 4 = 5*38 + 4 = 194

As you can see, the first term is always 5^(n-1), but the second term is not always (n-1). However, if we factor out a 5, we get:

a(4) = 5*(5*7 + 3) + 4 = 5*5*7 + 5*3 + 4 = 5^2*7 + 5^1*3 + 5^0*4

Now, if we continue this pattern, we can see that the second term is always (n-1), but the first term is changing. Let's try to write this in a general form:

a(n) = 5^(n-1)*(n-1) + b

Where b is a constant term that we need to determine. To find b, we can use the initial condition a(1) = 1:

a(1) = 5^(1-1)*(1-1) + b = 0 + b = 1
b = 1

Therefore, the explicit formula for this recursive sequence is:

a(n) = 5^(n-1)*(n-1) + 1

I hope this helps! Keep up the good work with your problem solving.
 

Related to Guessing Explicit Formula for a Sequence

1. What is an explicit formula for a sequence?

An explicit formula for a sequence is a mathematical expression that can be used to find any term in the sequence without having to know the previous terms. It is also known as a closed-form formula.

2. How do I find the explicit formula for a sequence?

To find the explicit formula for a sequence, you need to first identify the pattern or rule that the sequence follows. This can be done by looking at the difference between consecutive terms or by finding a common ratio between terms. Once the pattern is identified, it can be written as a mathematical expression to create the explicit formula.

3. Can an explicit formula be used for any sequence?

No, not all sequences have an explicit formula. Some sequences may follow complex patterns that cannot be easily expressed as a mathematical formula. In these cases, alternative methods such as recursive formulas or graphs may be used to represent the sequence.

4. Are there different ways to write an explicit formula?

Yes, there can be different ways to write an explicit formula depending on the sequence. For example, a geometric sequence can be written as a^n, a(n+1), or a(n-1) + d, where a is the first term and d is the common ratio. It is important to choose the most efficient and accurate way to write the formula for a given sequence.

5. What are some common mistakes when guessing an explicit formula for a sequence?

One common mistake is assuming that a sequence is arithmetic or geometric when it is actually a different type of sequence. It is important to carefully analyze the sequence and its pattern to determine the correct formula. Another mistake is not accounting for any initial values or constants in the formula, which can lead to incorrect results.

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