# 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;        