# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q#3

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**Question**

The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49.

** i. **Find the first term of the progression and the common difference.

The nth term of the progression is 46.

** ii. **Find the value of n.

**Solution**

i.

** **

From the given information, we can compile following data for Arithmetic Progression (A.P);

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

So we can write 9^{th} term as;

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

So we can write the expression for the sum of 4 terms in the Arithmetic Progression (A.P) is:

We can simultaneously solve the two equations of 9^{th} term and sum of 4 terms;

We can multiply both sides of the equations of 9^{th} term and by 4;

Subtracting the two equations;

Now we can find the first term from any of the two equations above by substituting

We choose equation of 9^{th} term;

ii.

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

Since the nth term is 46.

From (i) we have;

Substituting these in the expression of the nth term.

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