Question

:: State ?(1), ?(2), and ?(3)
:: State the Inductive Hypothesis you would use if you were proving the statement was true for all positive

integers using mathematical induction
:: State the “Left Hand Side” of the statement in the ? = ? + 1 case

4. Σ'- (-3)' = 2**) 4. 2n+1+(-1)* 3:27 i=1

Answer 1

3: 860=1 ?- 94+39-__ _ + (-)"ima = 6in (hu) L48 = P(1)= 12 -1 I PW = $1972!1+2 = 1 = R HR PC)=12-22 = -3 || 9(2) = (-19224.4.3- 3 PCB) = -92+3° = 6 |1(39 = x2 - 6132-1.3.4.6 Hence verified for n= 1, 2, 3 Now pooy by mathematical indiction. Parodi (w=1°-22+30- -+(3^2 = 6419-' n (int) Step Lim venify for nt, f(n) is done LHS = P (I) = 1²1 RHS = (-1). 1.2 at THE RHS Honce verified for not Step2: Let for n=k, Plus is due. P (K) = 12 -2 ² +3 – – – – + (-1) kaly? =(-1* K. (+1) . 2 - 0 Note: Now, Ich we prove that plm is done her vakt), then it is done for all n. Step3: To prove slag is den for n=kt) LHS = P(X) = 1² – 2² +2² - – – +(-1* - ² + 1 - K + 1 - Air z

LH: 8(x+1)= Eink! K (K) & Cik Citr vom 0} = fjk" (k+) Fş - (ku)] = (-) + (xt) (-4-2) 11.15 = P(x+) = (-10% (kx) (x+2) RAD = P(x+1)= (-1948A (RH) (x+2) L1 = RHS Hoe na kt! - P(m) is dam for all & by mathematical 2 induction.

nul - ). ) + (-) . PEL) = +59 +49 = ++-3 | Phn) - Saint C 3.2m N- Note: given question is wrong. the value of from jo to verify LH = RH- should start 2+1 Mies: 698 +6245 + 65 311) - Ceny 16> (49+63 4:6274 674 M) Macopes X Now, prooy by mathematical induction Parol:- PCM) PC 2 = 5*** +(-)" stept: verity for nat Pin) is terme. LAS = P.6-1) = (-2) + (*** R = 1 (n=1) = 2!+'+(-)" 3.29 3.21 21

IL HS = RHS So, 1C1) is due for nal. Step2in het for n=k, f(n) is due. so, P(K)= dk (-1)} = 2 + + (-1)* 3.2* = K+1 Steps: Now, prove that f(n) is time for - P(ka) = geta (-29 = K (-4377 ( 3.2k 2 kt) 970 29 = (ra= 2*30.0*. Cy** * (Cy" 3.2* 2x+2+(-)k 2 2x+2 + (-10%.2 + (-)*** 3.2k+) LH5= pk + 1) = 2x+ 2 + (-1* {2 -3 24+2 + (-13 kt) 3.2 x+) RHS = P(x+1) = (2+ (-1)" - 24+2 + (-1) 4+1 L3.20 Inakt - 3.2 kt) Hence, I (n) is true also for nakt. for all &, PCn) is true by mathematical induction. 3.9kti Inakti

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