Question

write an explicit formula for the sequence
4,1.5,-1,-3.5

Answer 1

@whpalmer4

Answer 2

How did I know that was going to happen? :-)

Answer 3

Is this an arithmetic or geometric sequence?

Answer 4

lol what is the difference

Answer 5

Arithmetic sequence is one where each term differs from the previous term by a fixed amount called the common difference.

Geometric sequence is one where each term is the previous term * a fixed amount, called the common ratio.

Arithmetic sequence example: 1, 2, 3, 4, 5, 6 ... each one is the previous one + 1
Geometric sequence example: 1, 2, 4, 8, 16, 32 ... each one is the previous one * 2

Answer 6

oh ok let me see

Answer 7

im not sure im stuck

Answer 8

Subtract the second term from the first term. Subtract the third term from the second term. Do you get the same answer each time?

Answer 9

4-1.5 = 2.5
1.5 - (-1) = 2.5
-1 - (-3.5) = 2.5
that's an arithmetic sequence.

The comparable test for a geometric sequence would be to divide instead of subtracting. For my 1,2,4,8,16 sequence:
2/1 = 2
4/2 = 2
8/4 = 2
etc.
that's a geometric sequence.

Answer 10

ok so it is arithmetic. what do we do next

Answer 11

Well, what is the common difference? What do we subtract from each term to get the next term?

Answer 12

Hello?

Answer 13

im sorry i got stuck watching the baby i got sidetracked sorry

Answer 14

oh and 2.5

Answer 15

so we need to write a formula for this sequence. does your book tell you the general formula for an arithmetic sequence?

Answer 16

Usually it is \[a_n = a_1 + (n-1)d\]where \(n\) is the number of the term, and \(d\) is the common difference. We need to use \(d=-2.5\) because each term gets smaller, not larger.

What will we use for the value of \(a_1\)?

Answer 17

\(a_1\) is just the first term in the sequence.

Answer 18

The sequence is 4, 1.5, -1, -3.5, ...

\[a_1= 4\]\[d = -2.5\]Plug those values into the formula and you have your equation for the sequence.

Answer 19

Test it out by finding the value of \(a_5\) for me.

Answer 20

and \(a_{100}\) also! :-)

Answer 21

um i dont know how what do i put in for n

Answer 22

@whpalmer4 im back i will not leave you this time promise. =D

Answer 23

As I said earlier, \(n\) is the number of the term. You're writing a formula, so you won't know the values of everything necessarily.

When you are computing \(a_{100}\) you'll use \(n = 100\).

Answer 24

thats what i thought but when i put a1=4, d=-2.5, and n=5 then answer came out 5.5???

Answer 25

is it right???

Answer 26

Show me your work...

Answer 27

\[4+(5-1)-2.5=5.5\]
thats what i get when i uses my calculator

Answer 28

\[a_n=a_1+(n−1)d\]
\[a_n = 4 + (5-1)(-2.5) =\]

Answer 29

You added -2.5 instead of multiplying by it, I bet! That's what you wrote here, at least...

Answer 30

ok i see what i did
it should be
\[4+(5-1)(-2.5)=-6\]

Answer 31

Right. So what does \(a_{100}=\)

Answer 32

=-243.5

Answer 33

@whpalmer4 thank you for your help!!

Answer 34

Sorry, fat-fingered the calculation myself, -243.5 is correct :-)

Answer 35

oh lol i was like what o.0

Answer 36

If you need a handy table of the first 100 terms, here you go :-)

\[\begin{array}{cc}
1. & 4. \\
2. & 1.5 \\
3. & -1. \\
4. & -3.5 \\
5. & -6. \\
6. & -8.5 \\
7. & -11. \\
8. & -13.5 \\
9. & -16. \\
10. & -18.5 \\
11. & -21. \\
12. & -23.5 \\
13. & -26. \\
14. & -28.5 \\
15. & -31. \\
16. & -33.5 \\
17. & -36. \\
18. & -38.5 \\
19. & -41. \\
20. & -43.5 \\
21. & -46. \\
22. & -48.5 \\
23. & -51. \\
24. & -53.5 \\
25. & -56. \\
26. & -58.5 \\
27. & -61. \\
28. & -63.5 \\
29. & -66. \\
30. & -68.5 \\
31. & -71. \\
32. & -73.5 \\
33. & -76. \\
34. & -78.5 \\
35. & -81. \\
36. & -83.5 \\
37. & -86. \\
38. & -88.5 \\
39. & -91. \\
40. & -93.5 \\
41. & -96. \\
42. & -98.5 \\
43. & -101. \\
44. & -103.5 \\
45. & -106. \\
46. & -108.5 \\
47. & -111. \\
48. & -113.5 \\
49. & -116. \\
50. & -118.5 \\
51. & -121. \\
52. & -123.5 \\
53. & -126. \\
54. & -128.5 \\
55. & -131. \\
56. & -133.5 \\
57. & -136. \\
58. & -138.5 \\
59. & -141. \\
60. & -143.5 \\
61. & -146. \\
62. & -148.5 \\
63. & -151. \\
64. & -153.5 \\
65. & -156. \\
66. & -158.5 \\
67. & -161. \\
68. & -163.5 \\
69. & -166. \\
70. & -168.5 \\
71. & -171. \\
72. & -173.5 \\
73. & -176. \\
74. & -178.5 \\
75. & -181. \\
76. & -183.5 \\
77. & -186. \\
78. & -188.5 \\
79. & -191. \\
80. & -193.5 \\
81. & -196. \\
82. & -198.5 \\
83. & -201. \\
84. & -203.5 \\
85. & -206. \\
86. & -208.5 \\
87. & -211. \\
88. & -213.5 \\
89. & -216. \\
90. & -218.5 \\
91. & -221. \\
92. & -223.5 \\
93. & -226. \\
94. & -228.5 \\
95. & -231. \\
96. & -233.5 \\
97. & -236. \\
98. & -238.5 \\
99. & -241. \\
100. & -243.5 \\
\end{array}\]

Answer 37

If you need any other values, well, you know the formula :-)

Answer 38

thanks again