The correct answers are:
(1) 6, 11, 16, 21, 26 (Option B)
(2)[tex]~a_1 = 8 [/tex]; [tex]~a_n = a_{n-1} - 2~[/tex] (Option A)
(3)[tex]~a_n = -2 + 3(n-1)~~~~[/tex] (Option D)
Explanations:
(1) Given Sequence:
[tex] a_n = 5n + 1 [/tex]
Now in order to find the first 5 terms, we need to put n=1,2,3,4,5 in the above sequence and solve.
For n=1: [tex] a_1 = 5(1) + 1 = 6[/tex]
For n=2: [tex] a_2 = 5(2) + 1 = 11[/tex]
For n=3: [tex] a_3 = 5(3) + 1 = 16[/tex]
For n=4: [tex] a_4 = 5(4) + 1 = 21[/tex]
For n=5: [tex] a_5 = 5(5) + 1 = 26[/tex]
Hence, the first five terms are: 6, 11, 16, 21, 26 (Option B)
(2) Given Sequence:
8, 6, 4, 2, …
Now to find the recursive definition, we need to adopt trial-and-error approach.
As, [tex] a_1 = 8 [/tex] (meaning the first element of the sequence is 8), the second or nth value of the sequence can be found by using the following formula:
[tex] a_n = a_{n-1} - d [/tex] --- (1)
Where, n = the index of the number in a sequence
d = difference between two consecutive numbers = 8-6 = 2
Now,
The second number of the sequence has to be 6 by using (1). Put n = 2 and d = 2 in (1):
[tex] a_2 = a_{2-1} - 2 [/tex]
[tex] a_2 = a_{1} - 2 [/tex]
Since [tex] a_1 = 8 [/tex], therefore,
[tex] a_2 = 8 - 2 = 6 [/tex] (correct)
Hence the correct answer is [tex] a_1 = 8 [/tex]; [tex]~a_n = a_{n-1} - 2~[/tex] (Option A)
(3) Given Sequence:
−2, 1, 4, 7, …
To find the explicit definition, use the following formula:
[tex] a_n = a_1 + (n-1)*d [/tex] --- (X)
Where,
[tex] a_n = nth~term~of~the~sequence \\a_1 = 1st~term~of~the~sequence = -2 \\d = common~difference = 4-1 = 7-4 = 3 \\n = index~of~a~number~in~a~sequence \\ [/tex]
Plug in the values in (X):
(X)=> [tex] a_n = -2 + (n-1)*3~~~~[/tex] (Option D)
