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. Thankfully, you can convert an iterative formula to an explicit formula for arithmetic sequences. The recursive definition for the geometric sequence with initial term a and common ratio r is an an r a0 a. 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." Then he explores equivalent forms the explicit formula and finds the corresponding recursive formula. 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)." Explicit & recursive formulas for geometric sequences Google Classroom About Transcript Sal finds an explicit formula of a geometric sequence given the first few terms of the sequences. 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 Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. Substitute a and b into 2=a+b+c: 2=1+0+c, c=1 Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Then subtract the 2 equations just produced: Saying 'the nth term' means you can calculate the value in position n, allowing you to find any number in the sequence. Therefore, this is not the value of the term itself but instead the place it has in the geometric sequence. Solve this using any method, but i'll use elimination: The first term is always n1, the second term is n2, the third term is n3 and so on. The function is y=ax^2+bx+c, so plug in each point to solve for a, b, and c. 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)
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