![]() Thankfully, you can convert an iterative formula to an explicit formula for arithmetic sequences. We can find the closed formula like we did for the arithmetic progression. To get the next term we multiply the previous term by r. In the explicit formula "d(n-1)" means "the common difference times (n-1), where n is the integer ID of term's location in the sequence." The recursive definition for the geometric sequence with initial term a and common ratio r is an an r a0 a. In the iterative formula, "a(n-1)" means "the value of the (n-1)th term in the sequence", this is not "a times (n-1)." Even though they both find the same thing, they each work differently-they're NOT the same form. A + B(n-1) is the standard form because it gives us two useful pieces of information without needing to manipulate the formula (the starting term A, and the common difference B).Īn explicit formula isn't another name for an iterative formula. ![]() M + Bn and A + B(n-1) are both equivalent explicit formulas for arithmetic sequences. So the equation becomes y=1x^2+0x+1, or y=x^2+1ītw you can check (4,17) to make sure it's right Substitute a and b into 2=a+b+c: 2=1+0+c, c=1 Then subtract the 2 equations just produced: Solve this using any method, but i'll use elimination: Improve your math knowledge with free questions in Convert between explicit and recursive formulas: geometric sequences and thousands of other math. The function is y=ax^2+bx+c, so plug in each point to solve for a, b, and c. This sequence is itself recursive, because the previous terms decide what the. Let x=the position of the term in the sequence Since the sequence is quadratic, you only need 3 terms. that means the sequence is quadratic/power of 2. However, you might notice that the differences of the differences between the numbers are equal (5-3=2, 7-5=2). This isn't an arithmetic ("linear") sequence because the differences between the numbers are different (5-2=3, 10-5=5, 17-10=7) Fibonacci numbers are also strongly related to the golden ratio: Binets formula expresses the n-th Fibonacci number in terms of n and the golden ratio, and.
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