Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  MayJun  (P19709/13)  Q#4
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Question
The 1st, 3rd and 13th terms of an arithmetic progression are also the 1st, 2nd and 3rd terms respectively of a geometric progression. The first term of each progression is 3. Find the common difference of the arithmetic progression and the common ratio of the geometric progression.
Solution
From the given information, we can compile following data about Arithmetic Progression (A.P);
Expression for the general term in the Arithmetic Progression (A.P) is:
Hence the 3^{rd} term of Arithmetic Progression (A.P) is;
Similarly the 13^{th} term of Arithmetic Progression (A.P) is;
According to the given conditions;
Therefore, we have;
Expression for the general term in the Geometric Progression (G.P) is:
Hence the 2^{nd }term of Geometric Progression (G.P) is;
Similarly the 3^{rd} term of Geometric Progression (G.P) is;
Now we can utilize the given conditions to write;









We can use the equations to find the value of .
From we can find that;
Substituting this value of in other equation;
Now we have two options.









In any Geometric Progression (G.P) is not possible as it would never produce any term beyond 1^{st} term because in Geometric Progression (G.P) each next term is obtained by multiplying previous term with .
Now we can fin value of by substituting in equation;
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