Question

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