# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#7

**Question**

**a) **The third and fourth terms of a geometric progression are and respectively. Find the sum to infinity of the progression.

**b) **A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector, find the angle of the largest sector.

**Solution**

**a) **

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

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

For the given case;

Expression for the general term in the Geometric Progression (G.P) is:

For the 3^{rd} term we can write;

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

For the given case;

**b) **

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

Smallest angle will be the first term;

Largest angle will be the last (5^{th}) term;

Sum of all the angle-sectors in a circle is .

Expression for the general term in the Arithmetic Progression (A.P) is:

For the 5^{th} term we can write;

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

For the given case;

This is the smallest angle; we know that largest angle is four times of the smallest angle i.e.

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