Can you explain how to find explicit and recursive formula of geometric sequence given 2 terms?
23. a<4> = -12, a<5> = -6
ratio: -6/-12 = 1/2
explicit:
a<n> = ar^(n-1)
-6 = a(1/2)^4
-6 = a(0.37)
-0.010 = a???
a<n> = -0.010(1/2)^n-1
recursive:
a<n> = a<n-1>r
a<n> = -0.010(1/2) ???
find 8th term:
a<n>=ar^n-1
a<8> = -0.010(1/2)^7
a<8> = -0.010(.0078125)
a<8> =-0.00078125
24. a<5>=738, a<2>= 12
ratio: 738/12 = 256/8 = 32 (I divided by 8 because 3 terms skipped. 2^3 = 8. How do i know the correct number to divide by? if terms aren't consecutive?)
explicit:
a<n> = ar^(n-1)
12 = a(32)^1
12 = 32a
0.375=a
a<n>= 0.375(32)^n-1
recursive:
a<n> = a<n-1>r
a<n> = a<n-1>32
To find the explicit and recursive formulas of a geometric sequence given two terms, you need to know the value of two terms and the common ratio.
Let's look at the examples you provided:
23. Given a<4> = -12 and a<5> = -6, the ratio is calculated by dividing the second term by the first term: -6 / -12 = 1/2.
To find the explicit formula, you can use the formula a<n> = a * r^(n-1), where a is the first term and r is the common ratio. In this case, the first term is -12 and the ratio is 1/2.
So, we have -6 = -12 * (1/2)^(4-1).
Simplifying the equation, we get -6 = -12 * (1/8).
If we divide both sides by -12, we get -6 / -12 = (1/8).
Solving this, we find that -0.5 = (1/8).
Therefore, the explicit formula for this geometric sequence is a<n> = -0.5 * (1/2)^(n-1).
For the recursive formula, you can use the formula a<n> = a<n-1> * r, where a<n-1> is the previous term and r is the common ratio. In this case, the ratio is 1/2.
So, the recursive formula for this geometric sequence is a<n> = a<n-1> * (1/2).
To find the 8th term, you can substitute n = 8 into the explicit formula and calculate it. So, a<8> = -0.5 * (1/2)^(8-1).
Simplifying, we get a<8> = -0.5 * (1/2)^7.
Calculating this, we find a<8> = -0.5 * 0.0078125 = -0.00078125.
24. Given a<5> = 738 and a<2> = 12, you need to calculate the ratio. In this case, the ratio is calculated by dividing a<5> by a<2>: 738 / 12 = 61.5.
To find the explicit formula, we can use a<n> = a * r^(n-1), where a is the first term and r is the common ratio. In this case, the first term is 12 and the ratio is 61.5.
So, we have 738 = 12 * 61.5^(5-1).
Simplifying, we get 738 = 12 * 61.5^4.
If we divide both sides by 12, we get 738 / 12 = 61.5^4.
Solving this, we find that 61.5^4 ≈ 26244.
Therefore, the explicit formula for this geometric sequence is a<n> = 12 * (61.5)^(n-1).
For the recursive formula, we can use a<n> = a<n-1> * r, where a<n-1> is the previous term and r is the common ratio. In this case, the ratio is 61.5.
So, the recursive formula for this geometric sequence is a<n> = a<n-1> * 61.5.
To find the 8th term, you can substitute n = 8 into the explicit formula and calculate it. So, a<8> = 12 * (61.5)^(8-1).
Simplifying, we get a<8> = 12 * (61.5)^7.
Calculating this, we find a<8> ≈ 2359015314400.
Regarding your question about dividing by 8 in the second example, you can only divide by a specific number if you know there were consecutive terms skipped. In this case, you knew that there were 3 terms skipped between a<2> and a<5>, so you divided by 2^3 = 8. If the terms are not consecutive, you need to figure out the correct number of terms skipped in order to divide by the correct power of the ratio.
a<5> = 738
a<4> = a<5>/32 = 738/32 = 23.0625
a<3> = a<4>/32 = 23.0625/32 = 0.7207
a<2> = a<3>/32 = 0.7207/32 = 0.0225
So the recursive formula is:
a<n> = a<n-1>/32
To find the 8th term:
a<8> = a<7>/32 = (a<6>/32)/32 = ((a<5>/32)/32)/32 = (((738/32)/32)/32)/32 = 0.00013542175