Solution
- The question gives us the following arithmetic sequence:
[tex]t(n+1)=t(n)-4[/tex]- We are asked to write out the explicit function for the sequence.
- To do this, we simply write out the terms of the sequence. After this, we would determine the common difference of the sequence.
- We will use the common difference to find the first term of the sequence and then use the formula below to find the explicit form of the arithmetic sequence:
[tex]\begin{gathered} t(m)=a+(m-1)d \\ \text{where,} \\ a=\text{first term of the sequence} \\ d=\text{common difference} \\ m=\text{ number of terms} \\ \\ (\text{Note that: }m=n+1\text{, since the formula starts assumes that the sequence starts from the 1st term not zeroth term} \end{gathered}[/tex]- We have been given that the second term of the sequence is 10. We would also use this term to form our sequence.
[tex]\begin{gathered} \text{Let n=2} \\ \text{The formula becomes} \\ t(2+1)=t(2)-4 \\ t(3)=t(2)-4 \\ \\ \text{But we know that }t(2)=10 \\ \therefore t(3)=10-4 \\ t(3)=6 \\ \\ \text{Let n=3} \\ t(3+1)=t(3)-4 \\ t(4)=6-4 \\ t(4)=2 \\ \\ \text{Let n=4} \\ t(4+1)=t(4)-4 \\ t(5)=2-4 \\ t(5)=-2 \\ \\ \text{Thus, we can write out the terms of the sequence as follows:} \\ \ldots,t(2),t(3),t(4),t(5),\ldots=\ldots10,6,2,-2\ldots \\ \\ \text{ We can observe that the common difference is -4 since,} \\ 6-10=-4 \\ 2-6=-4 \\ -2-2=-4 \\ \text{And so on}\ldots \\ \\ \text{Thus, we can trace our sequence back to its first term as follows:} \\ t(2)-t(1)=-4 \\ 10-t(1)=-4 \\ \text{Subtract 10 from both sides} \\ -t(1)=-4-10 \\ \therefore t(1)=14 \\ \\ t(1)-t(0)=-4 \\ 14-t(0)=-4 \\ \text{Subtract 14 from both sides} \\ -t(0)=-4-14=-18 \\ \therefore t(0)=18. \\ \\ \text{Thus, the first term }t(0)=18 \end{gathered}[/tex]- Let us now apply the formula for the nth term of a sequence to find the explicit formula:
[tex]\begin{gathered} first\text{ term =}t(0)=a=18 \\ \text{common difference}=d=-4 \\ t(m)=a+(m-1)d \\ \\ t(m)=18+(m-1)(-4) \\ \text{Expand the bracket} \\ t(m)=18+m(-4)-1(-4) \\ t(m)=18-4m+4 \\ \\ \therefore t(m)=22-4m \\ \\ \text{Let us write the sequence in terms of n} \\ m=n+1 \\ t(n)=22-4(n+1) \\ t(n)=22-4n-4 \\ t(n)=18-4n \\ \\ \text{Thus, the explicit function is:} \\ t(n)=18-4n \end{gathered}[/tex]- With the above formula, we can proceed to populate the table. Let us use the formula to calculate all the terms for each value of n.
[tex]\begin{gathered} t(n)=18-4n \\ \\ \text{when n = 0} \\ t(0)=18-4(0) \\ t(0)=18 \\ \\ \text{when n= 1} \\ t(1)=18-4(1) \\ t(1)=14 \\ \\ \text{when n=2} \\ t(2)=18-4(2) \\ t(2)=10 \\ \\ \text{when n = 3} \\ t(3)=18-4(3) \\ t(3)=18-12=6 \\ \\ \text{when n = 4} \\ t(4)=18-4(4) \\ t(4)=2 \\ \\ \text{when n = 5} \\ t(5)=18-4(5) \\ t(5)=-2 \end{gathered}[/tex]- On the table, we have the values filled in below:
Final Answer
The explicit form of the sequence is:
[tex]t(n)=18-4n[/tex]
