# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#4

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 3rd term of Arithmetic Progression (A.P) is;

Similarly the 13th 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 2nd term of Geometric Progression (G.P) is;

Similarly the 3rd 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 1st 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;