Find an explicit rule for the nth term of the sequence.7, -7, 7, -7, ... (5 points)Select one:a. an = 7 • (-1)n-1b. an = 7 • (-1)nc. an = 7 • 1n-1d. an = 7 • 1n+1
Accepted Solution
A:
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 = 7Hence, 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 = -7Hence, 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 = -7Hence, 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].