# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/13) | Q#9

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Question

The first, second and third terms of a geometric progression are ,  and  respectively.

(i)       Show that  satisfies the equation 7k2 48k + 36 = 0.

(i)       Find, showing all necessary working, the exact values of the common ratio corresponding to  each of the possible values of k.

(ii)        One of these ratios gives a progression which is convergent. Find the sum to infinity.

Solution

i.

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

Hence;

ii.

We are required to solve the following equation obtained in (i);

Now we have two options.

Now we can find using these values of and following equation;

iii.

We are required to find the sum to infinity of a progression which is convergent.

A geometric series is said to be convergent if  (or   ).

Therefore, for the given case;

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

Therefore;