# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01R) | Year 2013 | June | Q#6

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;