Can you check my homework please?
Given the following representations, tell whether it's arithmetic/geometric, find the difference/ratio, find the explicit formula, find the recursive formula, and the given term. (This whole question was in the form of a table)
1. -6, 12, 24....
Type:geometric
Rati0: -2
explicit form: a<n> = -6(-2)^n-1.
(not sure about notation. trying to write -6 times -2 raised to n-1)
recursive form: a<n> = -2(a<n-1>), and a<1> = -6
(trying to write -2 times a sub n-1)
I'm not sure which formula to use to find given term of a<10>.
- using explicit: (here's my try)
a<10>= -6(-2)^10-1
a<10>= -6(-2)^9
a<10>= -6(-512)
a<10>= 3072
- using recursive: I'm not sure how to do it. but here's my reasoning. need to find previous value, a<9>. so if a<10> = 3062 and r = -2; then a<9> = 3072/-2 = -1536. Use a<9> as previous value in recursive formula.
a<10> = r(a<n-1>)
a<10> = -2(-1536)
a<10> = 3072
Is there an easier way to figure it out?
2. 10, 20, 30, 40...
type: arithmetic
difference: 10
explicit form: a<n> = 10n
recursive form: a<n> = a<n-1> +10, and a<1> = 10
given term of a<32>:
- using explicit:
a<32>= 10 + (n-1)10
a<32>= 10 + (32-1)10
a<32>= 10 + 31(10)
a<32>= 10 + 310 = 320
or do I use formula i just solved for? a<n> = 10n
a<32> = 10(32)= 320
- using recursive: my reasoning is that I need to find a<31> which is previous value. so if a<32> = 320 and difference = 10; then a<31> = 320 - 10 = 310. Use a<31> in recursive form.
a<32> = a<n-1> + d
a<32> = 310 + 10
a<32> = 320
3. -10, -8, -6, -4...
Type: arithmetic
difference: 2
Explicit form: a<n> = 2n - 12
Recursive form: a<n> = a<n-1> + 2, and a<1> = -10
given term of a<56>:
- using explicit:
a<56>= -10 + (n-1)2
a<56>= -10 + (56-1)2
a<56>= -10 + (55)2
a<56>= -10 + 110 = 100
or a<56> = 2n - 12
a<56> = 2(56) - 12
a<56> = 112 - 12 = 100 ??
- using recursive: need to find a<55> which is previous value. so if a<56> = 100 and difference = 2; then a<55> = 100 - 2 = 98. Use a<55> in recursive form.
a<56> = a<n-1> + d
a<56> = 98 + 2 = 100 ??
4. 72, 48, 32...
Type: geometric
ratio: .67
explicit form: a<n>= 72(.67)^n-1
recursive form: a<n>= .67(a<n-1>) and a<1> = 72
given term of a<5>:
- using explicit:
a<5>= 72(.67)^5-1
a<5>= 72(.67)^4
a<5>= 72(.20)
a<5>= 14.5
-using recursive:
need to find a<4> which is previous value. so if a<5> = 14.5 and ratio = .67; then a<4> = 14.5/.67 = 21.64. Use a<4> in recursive form.
a<5>= r(a<n-1>)
a<5>= .67(21.64) = 14.51
Questions:
1. When the problem asks for finding a certain term, do I use the formula I just came up with or do I use the standard form depending on the type of sequence?
2. Is there an easier way to find a term using the recursive formula?
If you have an explicit formula, coming up with a particular term is easy. If not, you have to use the recursion several times.
#1. try (-2)^(n-1)
#2. You can use the explicit formula, but showing how to handle it as a general arithmetic sequence is also nice.
Naturally, any arithmetic/geometric sequence has an explicit formula.
#3 same as #2
#4 ratio is 2/3. 0.67 is only an approximation. Unless decimal values are given, I'd avoid them.
1. When the problem asks for finding a certain term, you can use either the formula you just came up with or the standard formula depending on the type of sequence. Both approaches will give you the correct answer.
2. There isn't necessarily an easier way to find a term using the recursive formula, as it requires calculating each previous term. However, you can simplify your calculations by organizing them in a table or using a spreadsheet. This way, you can easily track the values and avoid mistakes during calculations. Additionally, with practice, you might develop patterns or techniques that make recursive calculations faster and more efficient.