Question

A sequence  is defined by

  ,

Where  is a constant, .

a)   Write down an expression for  in terms of k.

b)  Show that

Given also that

c)   Calculate the value of k.

d)  Hence, find the value of .

Solution

a.    

We are given that sequence  is defined by

 

We are required to find .

We can utilize the given expression for general terms beyond first term as;

We are given that ;

b.    

We are required to find .

We can utilize the given expression for general terms beyond first term as;

We have found in (a) that ;

c.    

We are given that;

We have found in (b) that;

Therefore;

Now we have two options.

We are given that  , therefore;

d.    

We are given that sequence  is defined by

 

We have found in (c) that;

Hence;

 

Similarly;

 

Moreover,

 

We can see that the sequence consists of successive 1 and  repetitively.

Therefore, in first 100 terms of the sequence there will be 50 times 1 and 50 times .

Hence, sum of all these 100 terms will be;