what is the recursive formula for this sequence?
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Recursive Sequence. A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given. If you know the nth term of an arithmetic sequence and you know the common difference , d , you can find the (n+1)th term using the recursive formula an+1=an+d .
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The running time of insertion sort is a constant times the number of inversions. Let I ( i ) I(i) I(i) denote the number of j < i j < i j<i such that A [ j ] > A [ i ] A[j] > A[i] A[j]>A[i]. Then ∑ i = 1 n I ( i ) \sum_{i = 1}^n I(i) ∑i=1nI(i) equals the number of inversions in A.
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Every time you need to find the nth term formula for a quadratic sequence, the method is the same. To find a n 2 + b n + c an^2+bn+c an2+bn+c, the steps are: Find the difference between each term, and find the second differences (i.e. the differences between the differences);
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there is an obvious pattern. Such sequences can be expressed in terms of the nth term of the sequence. In this case, the nth term = 2n. To find the 1st term, put n = 1 into the formula, to find the 4th term, replace the n's by 4's: 4th term = 2 × 4 = 8.
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To find the binomial coefficients for (a + b)n, use the nth row and always start with the beginning. For instance, the binomial coefficients for (a + b)5 are 1, 5, 10, 10, 5, and 1 — in that order. as “n choose r.” You usually can find a button for combinations on a calculator.
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The formula for combinations is nCr = n! / r! * (n - r)!, where n represents the number of items, and r represents the number of items being chosen at a time. John is selecting three toppings from the eight offered by Pizza King.
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A recursive formula designates the starting term, a1, and the nth term of the sequence, an , as an expression containing the previous term (the term before it), an-1. The process of recursion can be thought of as climbing a ladder.
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The formula for combinations is nCr = n! / r! * (n - r)!, where n represents the number of items, and r represents the number of items being chosen at a time. John is selecting three toppings from the eight offered by Pizza King. 8 would represent our n term, and 3 would represent our r term.
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The number of combinations of n objects taken r at a time is determined by the following formula: C(n,r)=n! (n−r)! r!
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1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, ... (sequence A000108 in the OEIS).
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The nth partial sum of this series is sn = ( 1 - 1 2 ) + (1 2 - 1 3 ) + (1 3 - 1 4 ) + (1 4 - 1 5 ) + ··· + (1 n - 1 n + 1 ) = 1 - 1 n + 1 . sn = 1, the series converges with sum 1. arn = a + ar + ar2 + ar3 + ··· , where a is a constant and r is called the ratio of the series.
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A General Note: Recursive Formula A recursive formula is a formula that defines each term of a sequence using preceding term(s). Recursive formulas must always state the initial term, or terms, of the sequence.
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['a2 – b2 = (a – b)(a + b)', '(a+b)2 = a2 + 2ab + b2', 'a2 + b2 = (a + b)2 – 2ab.', '(a – b)2 = a2 – 2ab + b2', '(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.', '(a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca.', '(a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)']
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There is no general formula for n th derivation of any function, but say the function is e^x then the nth derivative also shall be e^x, for the function x^r the n th derivative shall be r*(r-1)*(r-2)*….. (r-n+1)*x^(r-n) and so on.
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Total = [ Compound interest for principal ] + [ Future value of a series ] Total = [ P(1+r/n)^(nt) ] + [ PMT × (((1 + r/n)^(nt) - 1) / (r/n)) ] Total = [ 5000 (1 + 0.05 / 12) ^ (12 × 10) ] + [ 100 × (((1 + 0.00416)^(12 × 10) - 1) / (0.00416)) ]
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A recursive formula designates the starting term, a1, and the nth term of the sequence, an , as an expression containing the previous term (the term before it), an-1. ... Find a recursive formula. This example is an arithmetic sequence (the same number, 5, is added to each term to get to the next term).
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So the formula a(n) = S(n) -S(n-1) works only for n > 1. For n = 1, a(n) = S(n) and that make sense because a(1) is first term and S(1) is sum of first 1 term.