Subjects algebra

Fibonacci 29Th Term 412852

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1. The problem is to find the 29th term in the Fibonacci sequence. 2. The Fibonacci sequence is defined by the recurrence relation: $$F_n = F_{n-1} + F_{n-2}$$ with initial terms: $$F_1 = 1, \quad F_2 = 1$$ 3. To find the 29th term, we can use the iterative method or the closed-form formula (Binet's formula). Here, we use the iterative method for clarity. 4. Starting from $F_1 = 1$ and $F_2 = 1$, calculate each term up to $F_{29}$: $F_3 = F_2 + F_1 = 1 + 1 = 2$ $F_4 = F_3 + F_2 = 2 + 1 = 3$ $F_5 = F_4 + F_3 = 3 + 2 = 5$ $F_6 = F_5 + F_4 = 5 + 3 = 8$ $F_7 = F_6 + F_5 = 8 + 5 = 13$ $F_8 = F_7 + F_6 = 13 + 8 = 21$ $F_9 = F_8 + F_7 = 21 + 13 = 34$ $F_{10} = F_9 + F_8 = 34 + 21 = 55$ $F_{11} = F_{10} + F_9 = 55 + 34 = 89$ $F_{12} = F_{11} + F_{10} = 89 + 55 = 144$ $F_{13} = F_{12} + F_{11} = 144 + 89 = 233$ $F_{14} = F_{13} + F_{12} = 233 + 144 = 377$ $F_{15} = F_{14} + F_{13} = 377 + 233 = 610$ $F_{16} = F_{15} + F_{14} = 610 + 377 = 987$ $F_{17} = F_{16} + F_{15} = 987 + 610 = 1597$ $F_{18} = F_{17} + F_{16} = 1597 + 987 = 2584$ $F_{19} = F_{18} + F_{17} = 2584 + 1597 = 4181$ $F_{20} = F_{19} + F_{18} = 4181 + 2584 = 6765$ $F_{21} = F_{20} + F_{19} = 6765 + 4181 = 10946$ $F_{22} = F_{21} + F_{20} = 10946 + 6765 = 17711$ $F_{23} = F_{22} + F_{21} = 17711 + 10946 = 28657$ $F_{24} = F_{23} + F_{22} = 28657 + 17711 = 46368$ $F_{25} = F_{24} + F_{23} = 46368 + 28657 = 75025$ $F_{26} = F_{25} + F_{24} = 75025 + 46368 = 121393$ $F_{27} = F_{26} + F_{25} = 121393 + 75025 = 196418$ $F_{28} = F_{27} + F_{26} = 196418 + 121393 = 317811$ $F_{29} = F_{28} + F_{27} = 317811 + 196418 = 514229$ 5. Therefore, the 29th term in the Fibonacci sequence is: $$\boxed{514229}$$