# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#9

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Question

a.   Two convergent geometric progressions, P and Q, have the same sum to infinity. The first and  second terms of P are 6 and 6r respectively. The first and second terms of Q are 12 and 12r  respectively. Find the value of the common sum to infinity.

b.   The first term of an arithmetic progression is  and the second term is , where  . The sum of the first 13 terms is 52. Find the possible values of .

Solution

a.

We are given two geometric progressions, P and Q, as;

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

 Expression for Common Ratio () in a Geometric Progression (G.P) is;

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

We are given that progressions, P and Q, have the same sum to infinity.

Now we can find the value of the common sum to infinity.

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

b.

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

Expression for difference  in Arithmetic Progression (A.P) is:

Therefore, for the given case;

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

Therefore, for the given case;

We have the trigonometric identity;

From this we can write;

Therefore;

Let ;

Now we have two options.

 Since ; Using calculator we can find ;

These are two possible values of  within the given interval .