Question

Question 1 10 pts A sequence d1, A2, A3, ... is defined by lettingai = 3 and ak = 70k-1 for all integers k > 2.Show that an = 3 .7h-1 for all integers n > 1. Tips and strategies: • Type many words to explain your reasoning. Use good punctuation. • To access the math editor, just click on the "square root of x" button in your toolkit. The button is indicated in the image below with a red box: B I V A - A - Ix 3 3 3 3 3 x x := = - NV Method V o 1 12pt When it is clicked, you will have two options: "basic view" and "advanced view." I recommend the basic view if you're not very experienced doing this. • Indicate where the proof starts by writing "Proof:" • Indicate where the proof ends by writing "QED" • Comments should be written (in brackets and in italic) whenever possible.

Answer 1

n-1 1-1 - 0 Proof 1 To prove an=3. 7 for all integers nal. Feist we prove this by the principle of Mathematical induction : step 2: When n=1 Given a = 3 = 3x1 = 3x7 - 307 So the result (an=3. f) is true for n=1, Assume Step 2: that the result is true for n=kah Fe = EXE = (k-D ata i assume akt = 3 E = e. F K -2 - 3 + assume T & cù, ax = 3.72 Step 3: To prove the result is true for n=k a = 79KY for k72 = 7.3.*** (substituting ()) Given na = 3.7 7K-2 K-241 - 3 3. Okay - true for n=k. Hence the result (a=3. 7) is Scanned with CamScanner

Thus by the principle of mathematical induction the result an=3.7 is true for all integers nol. CScanned with CamScanner

- 2x = 1*=== Proof 2 Given 9,= 3 = 3.7 ) to be Also given that are = fakt, for K7,2 when K=2, ag = 79, = 7*3 = 3.7' KE3, Q3 = 72 = 7.3.7' = 3.7 R=4, 94 = 72 = 7.3.7 = 3.73 K=5, 95= 724=7.3.7333.74 Hence in general an=3. 7 for all integers no ... anned with CamScanner