Exercises 63- Find the smallest n E N such that 2n^2 + 3n + 1 < n^3.
- Prove by induction for n > 0:
2+4+6+...+2n =n^2 +n - Prove by induction:
(a) 12 +2^2 +3^2 +...+n^2 =n(n+1)(2n+1)/6forn >0
(b) 13 +2^3 +3^3 +..+n^3 = (+2+3+...--+n)^2 forn >0
(c) 14 +2^4 +3^4 +... + n4 = n (n + 1) (2n + 1) (3n2 + 3n - 1)/30 for n > 0
(d) 15 +2^5 +3^5 +...+n^5 =ln^6 +an6 +n(^4) - nzforn>0
122 nfon
- Prove by induction:
(a) 0.2°1.2^01 +2.2^2 +3.2^3 +...+n.2n =(n-1)2n+1+2forn >0
(b) 12 +32 +52 + .. + ± (2n + 1)2 - (n + 1) (2n + 1) (2n + 3)/3 for n >^0
(c) 12 -2^2 +32 + ... + (-1)n-1 n^2 = (-1)n-' n (n + 1)/2 for n > 0
(d) 1.2±2.3+ 3.4+...+n.(n+ 1)=n(n+ 1)(n+2)/3forn >0
(e) 1.2.3+2.3.4+ 3.4.5+...+n.(n+ 1).(n+2)=n(n+ 1)(n+2)
(n + 3)/4 for n > 0- Prove by induction:
1 1 1
(b) 1 (a) 2 + 23 _ + .. + n (n-l) = 2 2 f- fornnl>-- - Prove by induction that 8 divides (2n + 1)2 - 1 for all n E N.
- Prove by induction for n > 0:
(a) 3 divides n^3 + 2n
(b) 5 divides n^5 - n
(c) 6 divides n^3 - n
(d) 6 divides n^3 + 5n - Prove by induction for all n E N:
(a) 7 divides n^7 - n
(b) 11 divides n 1 1 - n
(c) 13 divides n^13 - n
(d) 120 divides n^5 - 5n^3 + 4n - Prove by induction: The sum of the cubes of any three consecutive natural numbers is
divisible by 9. - Show that any integer consisting of 3n identical digits is divisible by 3n. Verify this for
222; 777; 222,222,222; and 555,555,555. Prove the general statement for all n e N by
induction. - Prove by induction that the following identities are true for the Fibonacci numbers:
(a) y 0 F2 i+1 = F2n+2 - 1 for n > 0
(b) y•n I Fi2 = Fn"- Fn,+l I for n _> I
(c) Fi=0 Fn+z-lforn>0- Find the Fibonacci numbers F 8 through F 15 .Prove the following results for the Fi-
bonacci numbers:
(a) F3n and F3n+l are odd, and F3n+2 is even for n > 0
(b) Fo + F 2 + ... + F 2 n = F 2 n+l for n > 0