Question

Use the two given terms to find an nth term rule for the sequence. a7 = -17 and a25 = -71

Answer 1

4-3n

Step-by-step explanation:

I assume this is supposed to be an AP.

use the general forms to find two simultaneous equations.

a+6d=-17

a+24d=-71

18d=-54

d=-3

a-18=-17

a=1.

so, we have the general term is 1-3(n-1) or 4-3n

Answer 2

The nth term rule for the sequence is
`a_n = -6n + 11.`

Step-by-step explanation:

To find the nth term rule for the sequence, we can start by identifying the common difference (d) between consecutive terms. The difference between a7 and a25 is
`(a25 - a7) / (25 - 7) = (-71 - (-17)) / (25 - 7) = -54 / 18 = -3.` Thus, the common difference is -3.

Now, we can use the formula for the nth term of an arithmetic sequence:
`a_n = a_1 + (n - 1)d`, where
`a_1` is the first term, n is the term number, and d is the common difference. We know that
`a_7 = -17,`so substituting these values into the formula gives us:
`-17 = a_1 + (7 - 1)(-3).` Solving for
`a_1, we get a_1 = -17 + 6(-3) = -17 - 18 = -35.`

Therefore, the nth term rule for the sequence is
`a_n = -35 - 3(n - 1).`Simplifying further, we get
`a_n = -6n + 11.`

In summary, the common difference allowed us to establish the arithmetic nature of the sequence, and using the arithmetic sequence formula with the given terms, we derived the nth term rule
`a_n = -6n + 11.`