The Limit of the Ratio of Successive Fibonacci Numbers 1. Restart and load the combinatorics library so we can access the Fibonacci function: 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 2. Create the sequence of the first 100 fibonacci numbers: LUkkc2VxRyUqcHJvdGVjdGVkRzYkNyRJIm5HNiItSSpmaWJvbmFjY2lHRig2I0YnL0YnOyIiIiIkKyI= 3. Look at the ratio of the fourth Fibonacci number to the third: 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 JSFH 4. Create the sequence of the ratios of successive fibonacci numbers up to n = 50: PkkpZmlicmF0aW9HNiI3Iy1JJHNlcUclKnByb3RlY3RlZEc2JDckSSJuR0YkKiYtSSpmaWJvbmFjY2lHRiQ2I0YrIiIiLUYuNiMsJkYrRjAhIiJGMEY0L0YrOyIiIyIjXQ== 5. Approximate this ratios to 20 decimal places: LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiRJKWZpYnJhdGlvRzYiIiM/ 6. Plot the sequence -- load plot library first so we can use pointplot: LUkld2l0aEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSZwbG90c0dGJA== LUkqcG9pbnRwbG90RzYiNiNJKWZpYnJhdGlvR0Yk 7. Evaluate 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 to 20 decimal places. LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiQtSSIqR0YkNiQsJiIiIkYqLUklc3FydEc2IjYjIiImRiojRioiIiMiIz8= 8. Verifying some of the golden ratio properties: PkkicEc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYkLUkiKkdGJzYkLCYiIiJGLS1JJXNxcnRHNiRGJ0koX3N5c2xpYkdGJDYjIiImRi0jRi0iIiMiIz8= QyQqJEkicEc2IiEiIiIiIg==LCZJInBHNiIiIiIhIiJGJQ== KiRJInBHNiIiIiM= Lucas Numbers: These are like the Fibonacci numbers, except the first two numbers are 1 and 3: PkkmbHVjYXNHNiJmKjYjSSJuR0YkRiQ2JEkpb3BlcmF0b3JHRiRJJmFycm93R0YkRiQsJi1GIzYjLCY5JCIiIiEiIkYwRjAtRiM2IywmRi9GMCEiI0YwRjBGJEYkRiQ= QyQ+LUkmbHVjYXNHNiI2IyIiIkYoRig=Pi1JJmx1Y2FzRzYiNiMiIiMiIiQ= LUkkc2VxRyUqcHJvdGVjdGVkRzYkNyRJIm5HNiItSSZsdWNhc0dGKDYjRicvRic7IiIiIiM1 PkkrbHVjYXNyYXRpb0c2IjcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkNyRJIm5HRiQqJi1JJmx1Y2FzR0YkNiNGKyIiIi1GLjYjLCZGK0YwISIiRjBGNC9GKzsiIiMiIz8= LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJK2x1Y2FzcmF0aW9HNiI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=