# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#8

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Question

a)   The first, second and last terms in an arithmetic progression are 56, 53 and 22 respectively.  Find the sum of all the terms in the progression.

b)   The first, second and third terms of a geometric progression are 2k + 6, 2k and k + 2  respectively, where k is a positive constant.

i.
Find the value of k.

ii.       Find the sum to infinity of the progression.

Solution

a)

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

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

For the given case;

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

For the given case;

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

For the given case;

b)

i.

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 we can have two options;

We can equate the above two expressions of ;

Now we have two options;

According to the given condition  is positive.

ii.

Substituting the value of ;

Also in;

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

For the given case;