Question

Part 1.] Indicate the general rule for the arithmetic sequence with $a_{3}=-4$ and $a_{8}=-29$
A.] $a_{n}=-6+(n-1)(-5)$
B.] $a_{n}=-6+(n-1)(5)$
C.] $a_{n}=6+(n-1)(-5)$
D.] $a_{n}=6+(n-1)(5)$

Part 2.] Which of the following is the general term for the sequence m, -m, m, -m, . . .?
A.] $m(-1)^{n-1}$
B.] $(-m)^{n}$
C.] $(-1)m^{n+1}$
D.] $(-1)m^{n-1}$

Part 3.] Indicate a general rule for the $n^{th}$ term of the sequence when $a_{1}=5$ and $r= \sqrt{3}$
A.] $a_{n}=( \sqrt{3})(5)^{n+1}$
B.] $a_{n}=( \sqrt{3})(5)^{n-1}$
C.] $a_{n}=(5)( \sqrt{3})^{n-1}$
D.] $a_{n}=(5)( \sqrt{3})^{n+1}$

Answer 1

These are 3 questions and 3 answers.

Part 1.] Indicate the general rule for the arithmetic sequence with A3 = - 4 and A8 = - 29

Answer: option C. An = 6 + (n-1)(-5)

Solution:

1) A3 is the third term
2) A8 is the eigth term
3) The formula for arithmetic sequences is: An = Ao + (n - 1)d

where n is the number of term and d is the difference between two consecutive terms.

=>

4) A8 = Ao + (8 - 1)d = - 29 => Ao + 7d = - 29  ----- [equation 1]

5) A3 = Ao + (3 - 1)d = - 4 => Ao + 2d = - 4 ------- [equation 2]

6) Subtract equation 2 from equation 1 => 7d - 2d = - 29 - (-4) =>

5d = - 29 + 4
5d = - 25
d = - 25/5 = - 5

7) Find Ao using equation 2:

 Ao + 2d = - 4 =>
 Ao = - 4 - 2d = - 4 - 2(- 5) = - 4 + 10 = 6

8) General rule: An = 6 + (n - 1) (-5) <-------- answer: option C.
 

Part 2.] Which of the following is the general term for the sequence m, -m, m, -m, . . .?

Answer: option a. m (-1)^ (n-1).

Justification:

the sign of the coefficient changes for each term.

when n = 1, the sign is positive: (-1)^ (1-1) = 1
when n = 2, the sign is  negative: (-1)^ (2-1) = - 1
when n = 3, the sign is positive: (-1)^ (3-1) = 2

And so on. So, m (-1)^ (n-1) does the work.

Part 3.] Indicate a general rule for the nth term of the sequence when A1 = 5 and r = √3


Answer: option C. An = (5)(√3)^(n-1)

Explanation:

This is a geometric sequence with A1 = 5 and r = √3

The terms of the geometric sequence are:

A1 = 5
A2 = A1 * √3 = 5√3
A3 = A2 * √3 = 5(√3)(√3) = 5(3) = 15
A4 = A3 * √3 = 15√3

So, the general expression is An = 5 * (√3)^(n-1), which is the option C.