Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#10

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Question

a)   An arithmetic progression contains 25 terms and the first term is 15. The sum of all the terms in  the progression is 525. Calculate

           i.       the common difference of the progression

        ii.       the last term in the progression

      iii.       the sum of all the positive terms in the progression. 

b)   A college agrees a sponsorship deal in which grants will be received each year for sports  equipment. This grant will be $4000 in 2012 and will increase by 5% each year. Calculate 

                    i.       the value of the grant in 2022

                   ii.       the total amount the college will receive in the years 2012 to 2022 inclusive.

Solution

a)
 

From the given information, we can compile following data about Arithmetic Progression (A.P); 


i.
 

Expression for the sum of  number of terms in the Arithmetic Progression (A.P) is:

Substituting the given values;


ii.
 

Expression for the general term  in the Arithmetic Progression (A.P) is:

For the given data;

From a(i) we have;

Therefore;


iii.
 

From the given data we can write the Arithmetic Progression (A.P) as;

Hence the first positive term in the Arithmetic Progression (A.P) is;

We know the last term in the Arithmetic Progression (A.P) from a(ii);

According to the given data there are total 25 terms in the Arithmetic Progression (A.P) and out of  these 6 are not positive (5 negative and one ZERO); therefore total number of positive terms is;

Expression for the sum of  number of terms in the Arithmetic Progression (A.P) is:

Hence for the positive terms of the Arithmetic Progression (A.P);

b)
 

From the given information, we can compile following data about second Geometric Progression  (G.P);

The amount on 2nd year;

The amount on 3rd year;

It is evident that it represents Geometric Progression (G.P);

Expression for Common Ratio () in a Geometric Progression (G.P) is;


i.
 

To find the amount in 2022 i.e. 11th term

Expression for the general term  in the Geometric Progression (G.P) is:

For the given data and ;


ii.
 

To find the total amount of grant;

Expression for the sum of  number of terms in the Geometric Progression (G.P) when  is:

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