# 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 2^{nd} year;

The amount on 3^{rd} 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. 11^{th} 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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