!!!!!algebra 2 question! need help ASAP!!!!!!

Question

1. Explain why we can have a converging infinite series when 0 < r < 1. Hint: Use "sum of a finite geometric series" as a basis for your explanation.                                                                                                                                                 
2. John begins a savings account as follows. On week 1, he puts 1 penny in his bank, on Week 2, he puts in 2 pennies, in Week 3 he puts in 3 pennies, etc. Write an equation that will model how much money he has saved after 'n' weeks and use it to find the amount of money that he will save in one year. How much money did he put in his bank during the final week? Show all work.

Answer 1

part 1:

 

Sum of finite geometric series has partial sum:

 

 Sn =  [1 - r^(n+1) ]/ (1-r)  where r is the common ratio

 

 

          Since  0 < r < 1,   

                    0 < r^2 < r < 1

                    0 < r^3 < r^2 < r < 1

                   ....

                    0 < r^(n) < r^(n-1) < r < 1  

 

Multiplying everything by 0<r<1,

                0 < r^(n+1) < r^n < ... <r^2 < r < 1

 

So the statement hold for any positive integer 0<N<infinity

 

          0  < r^(n) < 1 

 

which shows the common ratio converges to zero. The sum converges as a result

 

 

part 2:

 

   The sum IN PENNIES is  1 + 2 + 3 + 4 + .... + N which is the sum of the first N integers,where N is the # of weeks.

 

    It can also be shown by induction that the sum of these first N integers is

 

    N(N+1)/2 

 

 1 year is 52 weeks, so there will be

 

    52*53/2 = 26*53 = 1378

 

  $13.78