Find the indicated term of the Fibonacci sequence \( 15^{\text {th }} \) term

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... To find the 15th term of the Fibonacci sequence, we can use the formula for the nth term of the Fibonacci sequence: \[ F(n) = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}} \] where \( \phi \) is the golden ratio, approximately equal to 1.618033988749895. Substitute \( n = 15 \) into the formula to find the 15th term of the Fibonacci sequence. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\left(\frac{323606797749979}{200000000000000}\right)^{15}-\left(-\frac{323606797749979}{200000000000000}\right)^{-15}\right)}{\sqrt{5}}\) - step1: Remove the parentheses: \(\frac{\left(\frac{323606797749979}{200000000000000}\right)^{15}-\left(-\frac{323606797749979}{200000000000000}\right)^{-15}}{\sqrt{5}}\) - step2: Subtract the terms: \(\frac{\frac{323606797749979^{30}+200000000000000^{30}}{200000000000000^{15}\times 323606797749979^{15}}}{\sqrt{5}}\) - step3: Multiply by the reciprocal: \(\frac{323606797749979^{30}+200000000000000^{30}}{200000000000000^{15}\times 323606797749979^{15}}\times \frac{1}{\sqrt{5}}\) - step4: Multiply the fractions: \(\frac{323606797749979^{30}+200000000000000^{30}}{200000000000000^{15}\times 323606797749979^{15}\sqrt{5}}\) - step5: Multiply by the Conjugate: \(\frac{\left(323606797749979^{30}+200000000000000^{30}\right)\sqrt{5}}{200000000000000^{15}\times 323606797749979^{15}\sqrt{5}\times \sqrt{5}}\) - step6: Multiply the numbers: \(\frac{323606797749979^{30}\sqrt{5}+200000000000000^{30}\sqrt{5}}{200000000000000^{15}\times 323606797749979^{15}\sqrt{5}\times \sqrt{5}}\) - step7: Multiply the numbers: \(\frac{323606797749979^{30}\sqrt{5}+200000000000000^{30}\sqrt{5}}{5\times 200000000000000^{15}\times 323606797749979^{15}}\) The 15th term of the Fibonacci sequence is approximately 610.