SOLUTION: Please Help. Solve Problem by writing the first few terms of a sequence. anne borrows $1000 and agrees to pay $100 plus interest of 1% on the unpaid balance each month. find the p

Question 890214: Please Help. Solve Problem by writing the first few terms of a sequence. anne borrows $1000 and agrees to pay $100 plus interest of 1% on the unpaid balance each month. find the payments for the first six months and the remaining debt at the end of that period.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
if i understand this correctly, this is the sequence that will happen.

you borrow 1000
at the endof the first month, the remaining balance is equal to 1.01 * 1000 which is equal to 1010.
you pay 110 and the remaining balance is now 900.
at the end of the next month, the remaining balance is equal to 1.01 * 900 which is equal to 909.
you pay 109 and the remaining balance is now 800.
at the end of the nextg month, the remaining balance is equal to 1.01 * 800 which is equal to 808.
you pay 108 and the remaining balance is now 700.
this continues until the end of the loan.

Based on these calculations, the total payments after 6 months is equal to 645 dollars.

the actual sequence of payments is 110, 109, 108, 107, 106, 105, ...

the remaining balance after 6 months is equal to 400.

the actual sequence of the remaining balance is 900, 800, 700, 600, 500, 400, ...

these are both arithmetic sequences.

the formula for the nth term of an arithmetic sequence is An = A1 + (n-1)d

for the payments, that formula becomes An = 110 + (n-1) * (-1)

for the remaining balance, that formula becomes An = 900 + (n-1) * (-100)

the sum of the elements in an arithmetic sequence is equal to n * (A1 + An) / 2

for the 6th month, the numbers become:

payments:

A6 = 110 + (5) * (-1) = 110 - 5 = 105

Sum(An) from n = 1 to 6 = 6 * (110 + 105) / 2 = 6 * 215 / 2 = 3 * 215 = 645.

remaining balance:

A6 = 900 + (5) * (-100) = 900 - 500 = 400

sum of remaining balance doesn't apply to the remaining balance.

the payments and remaining balance for each time period of the loan are shown in the following picture of the excel spreadsheet used to make the calculations.

$$$

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