# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | June | Q#9

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Question

a.   Calculate the sum of all the even numbers from 2 to 100 inclusive,

2 + 4 + 6 + …… + 100

b.   In the arithmetic series

k + 2k + 3k + …… + 100

k is a positive integer and k is a factor of 100.

i.    Find, in terms of k, an expression for the number of terms in this series.

ii.    Show that the sum of this series is

50 + 5000/k

c.   Find, in terms of k, the 50th term of the arithmetic sequence

(2k + 1), (4k + 4), (6k + 7), …… ,

Solution

a.

We are required to calculate the sum of all the even numbers from 2 to 100 inclusive.

This sequence is given by;

2 + 4 + 6 + …… + 100

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

We need  and  or  and  to calculate this sum.

Let us utilize;

We can collect following data from this sequence.

We need to find .

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

We are given that  , therefore;

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

Hence;

Hence, there are 50 terms in this sequence.

Now we can calculate the sum;

b.

i.

We are given the arithmetic sequence;

k + 2k + 3k + …… + 100

We are required to find the number of terms in the series.

We can collect following data from this sequence.

We need to find .

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

We are given that  , therefore;

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

Hence;

ii.

We are required to calculate the sum of following sequence;

k + 2k + 3k + …… + 100

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

We need  and  or  and  to calculate this sum.

Let us utilize;

We can collect following data from this sequence.

We need to find . We have found in (b:i) that;

Now we can calculate the sum;

c.

We are given the arithmetic sequence;

(2k+1),  (4k+4), (6k+7)

We are required to find the 50th term in the series.

We can collect following data from this sequence.

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

Therefore;

We need to find d.

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

Hence;