Question

Problem 8: (i) Use the Principle of Mathematical Induction to prove that 2n+1(-1)" + 1 1 – 2 + 22 – 23 + ... + (-1)22" = for all positive integers n. (ii) Use the Principle of Mathematical Induction to prove that np > n2 + 3 for all n > 2.

Answer 1

problem-8: to prove O ose princple of mathematical nsuchion that 1-272²-23 + .. + (420 = mtl (+1+1 for all positive integers nou prool: Let pin) be the statement I 1-2 +22-23 + AGD2n = 2 vor EN for n=1, 1-2=- and 21+ () +1 +1 Pyt B so P() is tove. lch us assume that. PLK) is towe To prove PCKH) is tove since PCK) is true that 1-2 +2² - 23 + + (-1) k k = 2 kt) (-okt 1 To prove P(K) istove, gie, to prove 1-2 +2²-23 + +C-1 K & Lykth kt) = 2Kt2 cukt Now, 1-2 +2 ² - 23 t o +Cukk (Dkt 2 kt) kt (Hk + 1 - 2 U (tky 3

k+ 1 + 3x3 - 2kH (-10k + +(2+1) (kt) kto 3 = 2 kt cook to cesta kit2 + (-1, ktl , ktl +1 = cookt, kt2 + 2 kt (ko - rok 26th + 1 - (+) K+, K+2+1 2 Hence 1-2 +2²+ - +ch, ktl 2 lat) - 2k12 441 41 prkt) is the Then by induction method pin) is tave for all nen. gey 1-2 +22-23 + - T +CD2 = 2nt (adh +11 bor an positive integers n. 1 To prove 34 > 2+3 bor all n22 - 0 prools: vt pln be the statement ce n3 = n2+3 do now, (2)² = 8 22+3= 7 AS 877 -> (23 > 2²+3 - P(2) is the let us assume that PCK) is true tor some KZ2 7. ks > K2+3 - ? To prove P(KH) is tove, gie., to prove (x+3 > K+1) +3.

(by ) BUT CH3 = x + 3x² + 3 kt 1 > (2+3) + 3k² takt = 4x² + 3x + 4 7 2k² +2Kity = (K² + 2k to) +3. = (kH)2 +3. = (KiB > (k+1, ²+3. - P(KH) is true Then by induction method Pen) all nEN g.l. 3 > 0² + bor all n22. is true bor