Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2005  June  Q#9
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Question
An arithmetic series has first term and common difference .
a) Prove that the sum of the first n terms of the series is
Sean repays a loan over a period of months. His monthly repayments form an arithmetic sequence.
He repays £149 in the first month, £147 in the second month, £145 in the third month, and so on. He makes his final repayment in the nth month, where n > 21.
b) Find the amount Sean repays in the 21^{st} month.
Over the months, he repays a total of £5000.
c) Form an equation in , and show that your equation may be written as
d) Solve the equation in part (c).
e) State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.
Solution
a)
From the given information, we can compile following data about Arithmetic Progression (A.P);
For an arithmetic series;
We are required to add all the n terms of the arithmetic series.
We also write all the above terms in reverse order.
We add both sides of both these equations;
It can be seen that all the terms on Right Hand Side of the equation are same and appear n times in the equation, therefore;
b)
From the given information, we can compile following data about Arithmetic Progression (A.P);
There are n payments made and .
We are required to find the payment made in 21^{st} month.
Therefore, we are looking for 21^{st} term of this arithmetic series.
Expression for the general term in the Arithmetic Progression (A.P) is:
Hence;
We need to find .
Expression for difference in Arithmetic Progression (A.P) is:
From the given information we can find as follows;
Hence;
Hence, amount paid in 21^{st} payment is £109.
c)
We are given that if he makes n payment, he will repay a total of £5000.
As found in (a) sum of n terms of an arithmetic series is;
Substituting values of ;
d)
We are required to solve the equation;
It is evident that it is a quadratic equation.
For a quadratic equation , the expression for solution is;
Therefore;
Now we have two options.






e)
As shown above in (d), there may be either 50 or 100 total number of payments.
Let us consider both number of payments and find 50^{th} and 100^{th }payment amounts.
Expression for the general term in the Arithmetic Progression (A.P) is:
For n=50 
For n=100 










Since payment cannot be negative, n=100 is not sensible.
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