Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  MayJun  (P19709/11)  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 S_{n} is the sum of the first n terms.
i. Find an expression, in terms of p, q and n, for S_{n}.
ii. Given that S_{4}= 40 and S_{6}= 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 ;
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