Question

THE EXPLICIT FORMULA Use the following arithmetic sequence and the formula an= a1 + (n - 1)d to answer the questions below. 123, 116, 109, 102, 95, ... Part I: Find the value of each of the following: 21 = 123 and d=-17 Part II: Find the explicit formula. Show your work. Part III: Use the explicit formula you found in Part II to find the value of the 100th term in the sequence, a100. Show your work. State whether each sequence is arithmetic or geometric, and then find the explicit and recursive formulas for each sequence.

Answer 1

We have the sequence of numbers: 123, 116, 109, 102, 95,... and the formula:

`a_n=a_1+(n-1)d`

We need to find the value of a1 and d. We can see that the first value is a1=123, and the next values just substract 7 units from the previous, so d=-7. And this answer the Part 1.

For Part 2 we need to find the explicit formula, which already did:

`\begin{gathered} a_n=a_1+(n-1)d \\ a_n=123-7\cdot(n-1) \end{gathered}`

Part 3: Find the term of a100. We just need to replace n=100 in the formula above, so:

`a_(100)=123-7\cdot(100-1)=123-693=-570`

The sequence is arithmetic because,, as we already say above, the next value in the sequece add a constant from previuos value. In this case, the constant is d=-7, so the recursive formula is:

`a_n=a_(n-1)-7`