# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/11) | Q#8

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Question

a)   A geometric progression has a second term of 12 and a sum to infinity of 54. Find  the possible values of the first term of the progression.

b)  The nth term of a progression is p + qn, where p and q are constants, and Sn is  the sum of the first n terms.

i.       Find an expression, in terms of p, q and n, for Sn.

ii.       Given that S4= 40 and S6= 72, find the values of p and q.

Solution

a)

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

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

For the given case;

We need to find .

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

For the given case;

Hence;

Now we have two options.

 Since ;

b)

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

i.

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

Therefore, for the given case we already have but we need to find .

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

For the given case;

For we substitute ;

Hence;

ii.

We are given that;

As demonstrated in (b:i);

Therefore;

 For For

We can rearrange  as;

Substituting in ;

We can substitute in to find value of ;