Question

1. What are the first five terms of the sequence given by the formula an = 5n + 1?

1, 5, 9, 13, 17
6, 11, 16, 21, 26
5, 10, 15, 20, 25
5, 6, 7, 8, 9

2. Write a recursive definition for the sequence 8, 6, 4, 2, …
a1 = 8; an= an–1 − 2
a1 = 2; an= an–1 + 6
a1 = 8; an= an–1 + 2
a1 = 2; an= an–1 + 8
3. Write an explicit definition for the sequence −2, 1, 4, 7, …
an = –2 + 3n
an = –2 + (n – 3)
an = –2 + (n – 3)
an = –2 + 3 (n – 1)

Answer 1

1. a n = 5 n + 1
First 5 terms are:
B ) 6, 11, 16, 21, 26.
2. The sequence is : 8, 6, 4, 2, ...
A ) a 1 = 8; a n = a n-1 - 2
3. The sequence: - 2, 1, 4, 7,...
a 1 = - 2, common difference: d = 3
D ) a n = - 2 + 3 ( n - 1 )

Answer 2

The correct answers are:

(1) 6, 11, 16, 21, 26 (Option B)

(2)
`~a_1 = 8`;
`~a_n = a_(n-1) - 2~` (Option A)

(3)
`~a_n = -2 + 3(n-1)~~~~` (Option D)

Explanations:

(1) Given Sequence:

`a_n = 5n + 1`

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:
`a_1 = 5(1) + 1 = 6`

For n=2:
`a_2 = 5(2) + 1 = 11`

For n=3:
`a_3 = 5(3) + 1 = 16`

For n=4:
`a_4 = 5(4) + 1 = 21`

For n=5:
`a_5 = 5(5) + 1 = 26`

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,
`a_1 = 8` (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:

`a_n = a_(n-1) - d` --- (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):

`a_2 = a_(2-1) - 2`

`a_2 = a_(1) - 2`

Since
`a_1 = 8`, therefore,

`a_2 = 8 - 2 = 6` (correct)

Hence the correct answer is
`a_1 = 8`;
`~a_n = a_(n-1) - 2~` (Option A)

(3) Given Sequence:

−2, 1, 4, 7, …

To find the explicit definition, use the following formula:

`a_n = a_1 + (n-1)*d` --- (X)

Where,

`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 \\`

Plug in the values in (X):

(X)=>
`a_n = -2 + (n-1)*3~~~~` (Option D)