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.