Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#8

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Question

a)   A cyclist completes a long-distance charity event across Africa. The total distance is 3050 km.  He starts the event on May 1st and cycles 200 km on that day. On each subsequent day he reduces  the distance cycled by 5 km.

(i)          How far will he travel on May 15th?

(ii)        On what date will he finish the event?

b)  A geometric progression is such that the third term is 8 times the sixth term, and the sum of the  first six terms is . Find

(i)         the first term of the progression,

(ii)      the sum to infinity of the progression.

Solution

a)
 

From the given information, we can compile following data;


i.
 

The distance on first day;

The distance on 2nd day;

The distance on 3rd day;

We can sum up distances as;

It is evident that it represents Arithmetic Progression (A.P);

The expression for difference  in Arithmetic Progression (A.P) is:

For distance Arithmetic Progression;

Since we are required to find the distance covered on 15th Day;

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


ii.
 

It is evident that total distance will covered on the day when sum of distance covered day by day  becomes equal to total distance ie 3050 Km.

We need to find date  when .

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

It is evident that it is a quadratic equation and we need to solve it for .

Standard form of quadratic equation is;

Expression for solution of quadratic equation is;

For the given case;

Now we have two options.

61st cannot
be a date as we are looking for a date.

Hence, the distance will be covered on  20th May.

b)
 


i.
 

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

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

Therefore, we can write the 3rd and 6th terms as;

Substituting these expressions in the following equation;

We are given that sum of first six terms is .

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

Therefore;


ii.
 

Expression for the sum to infinity of the Geometric Progression (G.P) when  or ;

Therefore, for the given case;

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