jmc

Question

Let f_1, f_2, f_3, be a sequence of numbers such that f_n = f_n - 1 + f_n - 2 for every integer n 3. If f_7 = 83, what is the sum of the first 10 terms of the sequence?

Accepted Answer

Let's express each of the first 10 terms using only

f_{1}

and

f_{2}

:

f_{1} f_{2} f_{3} f_{4} f_{5} f_{6} f_{7} f_{8} f_{9} f_{10} = f_{1}, = f_{2}, = f_{1} + f_{2}, = f_{1} + 2 f_{2}, = 2 f_{1} + 3 f_{2}, = 3 f_{1} + 5 f_{2}, = 5 f_{1} + 8 f_{2}, = 8 f_{1} + 13 f_{2}, = 13 f_{1} + 21 f_{2}, = 21 f_{1} + 34 f_{2} .

(Notice anything interesting about the coefficients? These are the Fibonacci numbers!)

Adding both sides tells us that the sum of the first 10 terms is

55 f_{1} + 88 f_{2} = 11 (5 f_{1} + 8 f_{2}) = 11 f_{7} = 11 \cdot 83 = 913 .