Question

Write an explicit equation for the arithmetic sequence defined byt(n+1)= t(n)-4t(2) = 10

Answer 1

We need to find the explicit equation for the sequence:

`\begin{gathered} t\mleft(n+1\mright)=t\mleft(n\mright)-4 \\ \\ t(2)=10 \end{gathered}`

First, we can complete the given table. Notice that when we subtract 4 from the n-th term, we obtain the next term (n+1).

Then, to find a previous term, we can add 4. Thus. we obtain:

n t(n)

0 14+4 = 18

1 10+4 = 14

2 10

3 10-4 = 6

4 6-4 = 2

5 2-4 = -2

Now, observing the above relations, we need to write an expression for t(n) in terms of n:

n t(n)

0 18 = 18 - 0*4

1 14 = 18 - 1*4

2 10 = 18 - 2*4

3 6 = 18 - 3*4

4 2 = 18 - 4*4

5 -2 = 18 - 5*4

...

n 18 - n*4

Therefore, an explicit equation for the sequence is:

`t(n)=18-4n`