Question

Please help me with 1, 5 a. and b. , 6 and 7

EXERCISES 1.3 1. Use mathematical induction to prove that each of the following identities are valid for all € N. ..I +2 +3 + ... + = n(n+1) 6. I + 3 + 5 + ... + (2n - 1) = ? 1+2+...+ n (n+1)(2+1) d. P+2+...+ ((n + 1)]* e 2 + 2 + 2 + ... + 2" - 22" - 1) * For x,y E R. *** y**1 (x - y)(x* + x*-+ . . . +y") 2. Use mathematical induction to establish the following inequalities. *2. 2>n for all EN b. 2"> n for all n E Nn 25 *c.n! > 2 for all n EN n 24 d. P +2 +...+ <n for all E N R 2 2 e. 1+ 2+ ... + < w for all n E N. 23 3. Prove Theorem 1.3.4. 4. "Let S N N be defined by f(1) = 5. (2) = 13, and forn 22. (n) = 2/(n - 2) + (n - 1). Prove that f(n) = 3.2" + (-1)" for all n E N. S. For each of the following functions with domain N. determine a formula for f(n) and use mathematical induction to prove your conclusion 1 (1) and for n > 1.f(n) = (n − 1)f(n - 1) - b. f(1) = 1.f(2)= 4, and for n > 2.f(n) - 2 f(n-1)-f(n-2) + 2. c. f(1) = 1, and for a > 1. f(n) - * sin = 1). *d. f(1) - 1./(2) = 0, and forn > 2. f(n) (n - 2) e. For a, a, E R arbitrary, let f(1) = 4 (2) = 4z, and forn > 2. f(n) =- (= f(n - 2) (n-1) For 4,42 € R arbitrary, let (1) - S(2) - az, and forn > 2.f(n)-6 /n-2). 6. Lef:N+N be defined by f(1) = 1.5(2) = 2, and f(n + 2) = v(n + 1) + s(m). Use Theorem 1.3.4 to prove that I s f(n) s 2 for all E N. 7. *Prove that *+pl +...+- + 1,1EN without using mathematical induction.

Answer 1

solution 1] Parta) + 2 + 3 + . . . + n n an ) for po I= 1(1+1). so statement is true for Pck) 1+2+3+ ... +K = K (K ) statement is true for P(x) for plktı) 1+2+3+...+* ECK+!) - By adding (Ixis both side 1+2+3+ - +*+*+!) - KCK") ky K'+*+1 +26*2 - k****2 =(+ 1){ KA tence statement is truefort+1) Therefore pek+1) is true when Pck) is the Hence b mathematical Induction Pen) is true for every positive number n. 2 Pantb] +3+5+ .... n-1) =n2 for nal PCI) 1.12 so statement is true for P(1) for P(K) 1+3+5+ .. +2k-1) K2 statement is true for P(K) for pik+l) 1+3+5+ - +(2k-1) (2[*+)-1) = x +(2k +1) = K + 2K+ = (k+)) 2 Heno statement is tow for (K+1) when PCR) is true Hence by m. pen is tow for Every positive numberny

Solution of neneincant? Pene impost 2 149001 Cari), 6 so statement is true for Pri) Play ,272t .. tkr A(*+1) (2 K"}] statement is the Pixs Plat) 1²+z²+z²+...+6² + (x+² kck+1)(2k+1) + +)2 -(K+ ( K (2k +) + 46+1)) 6 Ek + 1) (k+1+1) (2(K+1) +1) I so statement is true Pli+1] - PCk) is thu Hence by mi Pen) is the for every positone numbern. when solutiond) 1*Det ... no. 14 inuenun? P(): 1 (2 68 (23) ²= ' statement is true for Pris for perc) 1322°r.. +x?! ( 17CK"] 2 a statement is true for P(x) for Pek+1) F t-- ++(k+))) = (**)) + (+)3 =(64)2 ( + +] so statement is true for Pekti) U*y* )2 = (**) extruy Hope it will help you