Find an explicit rule for the nth term of the sequence. 7, -7, 7, -7,

The first term is 7. The common ratio is -7/7 = -1. The general formula for the nth term of a geometric sequence can be used:
  an = a1·r^(n-1)
  an = 7·(-1)^(n-1)

(a) The ONLY explicit rule for the nth term of the sequence is [tex]a_n = 7 \times (-1)^{n-1}\\[/tex].

Step-by-step explanation:

Here, the given sequence is 7, -7, 7, -7, ...

The first term = 7, Second term = -7, Third term =-7 and so on..

Now check the given sequence for each given formula, we get:

(1)  [tex]a_n = 7 \times (-1)^{n-1}\\[/tex]

Now, for n = 1 : [tex]a_1 = 7 \times (-1)^{1-1}\\[/tex]

                                       [tex]= 7 \times (-1)^0 = 7 \times 1 = 7 \implies a_1 = 7[/tex]

Similarly, for, n = 2:  [tex]a_2 = 7 \times (-1)^{2-1}\\[/tex]

                                       [tex]= 7 \times (-1)^1 = 7 \times (-1) = -7 \implies a_2 = -7[/tex]

Hence, the given formula satisfies the given sequence.

(2)  [tex]a_n = 7 \times (-1)^{n}\\[/tex]

Now, for n = 1 : [tex]a_1 = 7 \times (-1)^{1}\\[/tex]

                                       [tex]= 7 \times (-1)^1 = 7 \times (-1) = -7 \implies a_1 = -7[/tex]

But, First term = 7

Hence, the given formula DO NOT satisfy the given sequence.

(3)  [tex]a_n = 7 \times (1)^{n-1}\\[/tex]

Now, for n = 1 : [tex]a_1 = 7 \times (1)^{1-1}\\[/tex]

                                       [tex]= 7 \times (1)^0 = 7 \times 1 = 7 \implies a_1 = 7[/tex]

Similarly, for, n = 2:  [tex]a_2 = 7 \times (1)^{2-1}\\[/tex]

                                       [tex]= 7 \times (1)^1 = 7 \times (1) = 7 \implies a_2 = 7[/tex]

But, Second term = -7

Hence, the given formula DO NOT satisfy the given sequence.

(4) [tex]a_n = 7 \times (1)^{n+1}\\[/tex]

Now, for n = 1 : [tex]a_1 = 7 \times (1)^{1+1}\\[/tex]

                                       [tex]= 7 \times (1)^2 = 7 \times 1 = 7 \implies a_1 = 7[/tex]

Similarly, for, n = 2:  [tex]a_2 = 7 \times (1)^{2+1}\\[/tex]

                                       [tex]= 7 \times (1)^3 = 7 \times (1) = 7 \implies a_2 = 7[/tex]

But, Second term = -7

Hence, the given formula DO NOT satisfy the given sequence.

So, the ONLY explicit rule for the nth term of the sequence is [tex]a_n = 7 \times (-1)^{n-1}\\[/tex].

Try [tex]a_n=7(-1)^{n+1}[/tex], where [tex]n\ge1[/tex].