Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  OctNov  (P19709/12)  Q#8
Question
a) A cyclist completes a longdistance 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 2^{nd} day;
The distance on 3^{rd} 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 15^{th} 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.






61^{st} cannot 
Hence, the distance will be covered on 20^{th} 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 3^{rd} and 6^{th} 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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