1.

Use the 3 step Mathematical Induction process to prove the following statements. You must label and show all three steps. Step 1: Prove n = 1 is true Step 2: Assume true for all n = k Step 3: plug in k+1 and show it is true. You choose any 3 to prove.

1. 12 + 32 + 52 +... + (2n - 1)2 = n(2n-1)(2n+1) 3 2. Prove n2 + n is always even. 3. Prove 32 - 1 is divisible by 8. 4. 36 + 324 + 900+...+(12n -6)2 = 12n(4n2 - 1) 5. 13 + 23 +33+... +n3 = z (Itu)u) 2Part 1 - Application The table below shows the February balance of a simple interest savings account each year from 2015 to 2021 Year 2015 2016 2017 2018 2019 2020 2021 Balance $12,000 $14,018 $16,036 $18,054 $20,072 $22,090 $24, 108 . Do the balances form an arithmetic or geometric sequence? . What is the d or the r ? . Write a formula for the balance in the account n years after February 2015. . Find the sum of the February balances from 2015 to 2032, inclusive.Part 2 - Find the error(s) and solve the problem correctly. Determine whether the sequence converges or diverges. How do you know if it converges or diverges? 1 + 5+ 25 + ... Answer: Converges because it has a sum and r > 1.Part 3 - Discussion Question State whether the following statement is true or false. Be sure to explain your reasoning. An Infinite Geometric Series will never have a sum because the list of numbers will never end so how can you add an infinite list of numbers.At the neighborhood playground, your friend sits on a swing and gets pulled back by you to the highest point that you can reach. When you let go, your friend rides the swing without receiving or providing additional push. Your friend travels along an arc of 14 feet on the first swing (back to front). As the swing comes backwards, the length of the arc is .80 of the previous swing. This continues: each successive arc created by swinging is .80 of the previous swing. a. Find the length of the arc for the 5th swing (the 5th arc). Round to the thousandths place. b. On which swing is the length of the arc first less than 1 foot? c. Find the total distance your friend travels in all of the swings up to and including the swing in answer to letter b.Suppose you received a job offer with a starting salary of $48,500 per year and a guaranteed raise of $1200 per year. Find the number of years n until you will have earned a total of $321,000 since starting this job.Bob has 8 pieces of square index cards that each measure 8 inches per side. Using a pair of scissors, he cuts each piece in half (so the resulting size is 8" x 4") and then places all the resulting card pieces in a new stack. Bob then marks the center along the long edge and cuts all pieces in half. If he repeats this procedure 4 more times, how many pieces of card stock will he have and what will be the measure of each piece? (Note: All cuts will be made along the longer edge of the piece if there is an edge that is longer than the other edges.)