Fibonacci series textbook "Number Theory" by George E. Andrews (1971 edition)

[jldawson45]

Trying to return to number theory after a gap of 55 years is hard work. It's made even harder by apparent errors in the exercises in my textbook (see image). And, typically, these are the examples to which no worked answer is given. Am I going doolally?

 

[Dr.Peterson]

Trying to return to number theory after a gap of 55 years is hard work. It's made even harder by apparent errors in the exercises in my textbook (see image). And, typically, these are the examples to which no worked answer is given. Am I going doolally?View attachment 36927

You're misinterpreting the notation; they are not being inconsistent.

In (11), the summation doesn't specify that each term has the form [imath]F_{2k-1}F_{2k}[/imath], but only that each term is [imath]F_{k}F_{k+1}[/imath] and they are summing over an odd number of terms, with k ranging from 1 to 2n-1 (the last being [imath]F_{2n-1}F_{2n}[/imath]).

Did you check the sums? You'll find that, for n=2, the sum is [math]F_1F_2+F_2F_3+F_3F_4=(1)(1)+(1)(2)+(2)(3)=1+3+6=9=3^2=F_4^2[/math] as claimed.

In (12), they are taking an even number of terms, with k ranging from 1 to 2n. For example, for n=2, the sum is [math]F_1F_2+F_2F_3+F_3F_4+F_4F_5=(1)(1)+(1)(2)+(2)(3)+(3)(5)=1+3+6+15=24=5^2-1=F_5^2-1[/math] as claimed.

 

[blamocur]

Proof by induction for (11) is pretty straightforward. I suspect (12) is the same.