# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2009 | January | Q#9

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Question

The first term of an arithmetic series is a and the common difference is d.

The 18th term of the series is 25 and the 21st term of the series is .

a.   Use this information to write down two equations for a and d.

b.   Show that a = –17.5 and find the value of d.

The sum of the first n terms of the series is 2750.

c.   Show that n is given by

d.   Hence find the value of n.

Solution

a.

We are given that point following data of Arithmetic Series;

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

Therefore, for both 18th and 21st terms we can write equations as;

 For 18th term; For 21st term;

b.

We have found two equations in terms of a and d in (a).

We can simultaneously solve these two equations.

We can write both equations for ‘a’ as follows;

Equating both equations;

Substituting  in any of the above two equations yields ‘a’;

c.

We are given that sum of first ‘n’ terms of arithmetic series is 2750.

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

Therefore;

Substituting values of ‘a’ and ‘d’ from (a) we get;

d.

We are required to find the value of ‘n’.

From (c) we have;

Now we have two options.

Since ‘n’ represents total number of terms being added, therefore, it cannot be negative. Hence;