# 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), …… ,

giving your answer in its simplest form.

**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 50^{th }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;

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