# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#9

Question

a.   The first term of a geometric progression in which all the terms are positive is 50. The third  term is 32. Find the sum to infinity of the progression.

b.   The first three terms of an arithmetic progression are ,  and  respectively, where x is an acute angle.

i.
Show that  .

ii.       Find the sum of the first twenty terms of the progression.

Solution

a.

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

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

For the 3rd term of Geometric Progression (G.P);

Since all the terms are positive we only .

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);

i.

Expression for difference  in Arithmetic Progression (A.P) is:

For the given case;

We know that ;

ii.

Since  is an acute angle, we can use calculator to find  from;

Therefore;

Expression for difference  in Arithmetic Progression (A.P) is:

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

For the given case, sum of first 20 terms, ;