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;