Question

Consider the sequence defined by b0 = 0, b1 = 1, b2 = 2, and bn = bn−1 + 2bn−2 − 2bn−3 for n ≥ 3. Determine the values of b3, b4, b5, b6, and a closed form expression for bn. (You will need to think about how to appropriately extend the type of closed form expression given above.)

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Answer to a math question Consider the sequence defined by b0 = 0, b1 = 1, b2 = 2, and bn = bn−1 + 2bn−2 − 2bn−3 for n ≥ 3. Determine the values of b3, b4, b5, b6, and a closed form expression for bn. (You will need to think about how to appropriately extend the type of closed form expression given above.)

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Santino
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111 Answers
To find the values of b3, b4, b5, b6, we can use the recursive formula bn = bn−1 + 2bn−2 − 2bn−3.

Let's start by finding b3:

b3 = b2 + 2b1 - 2b0
= 2 + 2(1) - 2(0)
= 2 + 2 - 0
= 4

Next, let's find b4:

b4 = b3 + 2b2 - 2b1
= 4 + 2(2) - 2(1)
= 4 + 4 - 2
= 6

Now, let's find b5:

b5 = b4 + 2b3 - 2b2
= 6 + 2(4) - 2(2)
= 6 + 8 - 4
= 10

Lastly, let's find b6:

b6 = b5 + 2b4 - 2b3
= 10 + 2(6) - 2(4)
= 10 + 12 - 8
= 14

The values of b3, b4, b5, b6 are 4, 6, 10, 14 respectively.

To find a closed form expression for bn, we can try to find a pattern in the sequence.

If we write out a few terms of the sequence, we have:

b0 = 0
b1 = 1
b2 = 2
b3 = 4
b4 = 6
b5 = 10
b6 = 14

It seems that the terms are increasing by multiples of 2: 1, 2, 4, 6, 10, 14. Let's examine the differences between the terms:

1 - 0 = 1
2 - 1 = 1
4 - 2 = 2
6 - 4 = 2
10 - 6 = 4
14 - 10 = 4

We can see that the differences are also increasing by multiples of 2: 1, 1, 2, 2, 4, 4. This suggests that the sequence can be represented by a quadratic function.

Let's try a quadratic function of the form bn = an^2 + bn-1, where a is a constant.

Substituting n=1 into the quadratic function, we have:

b1 = a(1)^2 + b0
1 = a(1) + 0
a = 1

So the quadratic function representing the sequence is:

bn = n^2 + bn-1

Now we can write the closed form expression for bn:

Answer: bn = n^2 + bn-1

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