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.
 

From the given information, we can collect following information about this Geometric Progression  (G.P).

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;

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