Arithmetic series word problem

Question

Answer this question please..

 

1. On his 12th birthday, Enoch's grandparents deposited $25 into a savings account for him. Each month after that up to and including his 20th birthday, they deposit $10 more than the previous month. How much money will Enoch have on his 20th birthday, excluding interest? Show your steps

Answer 1

a₁ = amount given on 12th birthday = 25

 

a₂ = amount given one month later = 35

 

a₃ = amount given two months later = 45 etc.

 

This is an arithmetic sequence with common difference d = 10

 

a_n = amount given in month n  

 

    = a₁ + (n-1)d = 25 + 10(n-1) = 10n + 15

 

Up to and including his 20th birthday, Enoch will have been given gifts for (8)(12)+1 = 97 months.

 

Total amount received = S₉₇ = sum of the first 97 terms

                                       

                                        = (97/2)[a₁ + a₉₇]

       

                                        = (97/2)[25 + 985] = $48985

 

Answer 2

The common difference between the deposits is $10.  The initial deposit is $25.

 

The arithmetic sequence is

 

D_t = D₁ + d(t - 1)

 

where:

D_t = deposit

D₁ = initial deposit

d = common difference

t = time (years)  ---> t=1  represents 12th birthday

 

D_t = 25 + 10(t - 1)

 

Evaluate D₉ to find the amount Enoch received on his 20th birthday.