Answer:
(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].
