# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#7

Question

a)   The first two terms of an arithmetic progression are 1 and respectively. Show that the sum  of the first ten terms can be expressed in the form , where a and b are constants to be  found.

b)  The first two terms of a geometric progression are 1 and respectively, where .

i.
Find the set of values of for which the progression is convergent.

ii.       Find the exact value of the sum to infinity when .

Solution

a)

From the given information, we can compile following data about Arithmetic Progression (A.P);   Expression for the sum of number of terms in the Arithmetic Progression (A.P) is: So we need to find the sum of first 10 terms of Arithmetic Progression (A.P).

The expression for difference in Arithmetic Progression (A.P) is: For the given case;   The trigonometric identity; We can have from this; Hence; Substituting the given values;     b)

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

Expression for Common Ratio ( ) in a Geometric Progression (G.P) is;     A geometric series is said to be convergent if (or ).

Therefore;           But for the given condition; We only consider; Hence; To find the sum to infinity of Geometric Progression (G.P);

Expression for the sum to infinity of the Geometric Progression (G.P) when (or ) is: From b(i); Since      Therefore;    