Solution
- The question gives us the following arithmetic sequence:
- 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:
![\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}]()
- We have been given that the second term of the sequence is 10. We would also use this term to form our sequence.
![\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}]()
- Let us now apply the formula for the nth term of a sequence to find the explicit formula:
- 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.
- On the table, we have the values filled in below:
Final Answer
The explicit form of the sequence is:
