By examining the following sequences, write a difference equation to represent t

Question

By examining the following sequences, write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence. a. {2, 4, 16, 256,...} b. {1, 2, 5, 11, 23,...} By examining the following sequences, write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence. a. {2, 4, 16, 256,...} b. {1, 2, 5, 11, 23,...} By examining the following sequences, write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence. a. {2, 4, 16, 256,...} b. {1, 2, 5, 11, 23,...}

Explanation / Answer

a. {2, 4, 16, 256,...}

We have first term a1= 2

second term a2 = 4 = (2)^2 = (a1)^2

third term a3 = 16 = (4)^2 = (a2)^2

fourth terma a4 = 256 = (16)^2 = (a3)^2

We can see from the above pattern that the next term is square of the previous term. Basis on this we can say that the nth term is square of (n-1)th term

Therefore an = (an-1)^2

= > an - (an-1)^2= 0

b. {1, 2, 5, 11, 23,...}

From the given pattern we can see that the next term is twice plus one of the previous term.

Hence we can write

an =   2an-1 +1

=> an -   2an-1 = 1

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