Show that the sequence 3, 6, 12, 24, … is a geometric sequence, and find the next three terms. This value is called the common ratio, r, which can be worked out by dividing one term by the previous term. In a geometric sequence, the term to term rule is to multiply or divide by the same value. The sequence will contain \(2n^2\), so use this: \ For example this is how the fibonacci seq is calculated. ![]() You can look into dynamic programming approach where u calculate F(I,N) just once and just reuse the value. ![]() Also rightly so you have identified that it will lead to an inefficient algorithm and stack overflow. The coefficient of \(n^2\) is half the second difference, which is 2. The formula for the nth term of the sequence is the one you have already mentioned. The common ratio is obtained by dividing the current. It is represented by the formula an a1 r (n-1), where a1 is the first term of the sequence, an is the nth term of the sequence, and r is the common ratio. Example 1: Find the 27 th term of the arithmetic sequence 5, 8, 11, 54 . A geometric sequence is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed number. The second difference is the same so the sequence is quadratic and will contain an \(n^2\) term. Given an arithmetic sequence with the first term a 1 and the common difference d, the n th (or general) term is given by a n a 1 ( n 1 ) d. Formulas of Arithmetic Sequence an nth term that has to be found a1 1st term in the sequence n Number of terms d Common difference Sn Sum of n. Work out the nth term of the sequence 5, 11, 21, 35. In this example, you need to add \(1\) to \(n^2\) to match the sequence. To work out the nth term of the sequence, write out the numbers in the sequence \(n^2\) and compare this sequence with the sequence in the question. Video Test 1 2 3 4 Finding the nth term - Worked example Question Find the nth term for this sequence: 1, 4, 7, 10. Example 1: Find the 6 th term in the geometric sequence 3, 12, 48. Half of 2 is 1, so the coefficient of \(n^2\) is 1. Finding the n th Term of a Geometric Sequence Given a geometric sequence with the first term a 1 and the common ratio r, the n th (or general) term is given by a n a 1 r n 1. In this example, the second difference is 2. The coefficient of \(n^2\) is always half of the second difference. ![]() The sequence is quadratic and will contain an \(n^2\) term. The first differences are not the same, so work out the second differences. Work out the first differences between the terms. Work out the nth term of the sequence 2, 5, 10, 17, 26. They can be identified by the fact that the differences in-between the terms are not equal, but the second differences between terms are equal. Quadratic sequences are sequences that include an \(n^2\) term. Finding the nth term of quadratic sequences - Higher
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