# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#10

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**Question**

**a.**In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five terms. The first term is .

**i.**Show that the common difference of the progression is .

**ii.**Given that the tenth term is 36 more than the fourth term, find the value of .

**b.**The sum to infinity of a geometric progression is 9 times the sum of the first four terms. Given that the first term is 12, find the value of the fifth term.

**Solution**

**a.
**

**i.
**

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

First, we find expression for sum of first ten terms of the Arithmetic Progression (A.P).

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

We are given the first term but we need the 10^{th} term.

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

From the given data, we can write expression for 10^{th }term of Arithmetic Progression (A.P).

Hence, expression for sum of first 10 terms of Arithmetic Progression (A.P):

Next, we find expression for sum of 11 to 15 terms of the Arithmetic Progression (A.P).

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

We need the 11^{th} and 15^{th} terms of Arithmetic Progression (A.P).

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

From the given data, we can write expression for 11^{th} term of Arithmetic Progression (A.P):

Similarly, from the given data, we can write expression for 11^{th} term of Arithmetic Progression (A.P).

Now we can find expression for sum of 11 to 15 terms of Arithmetic Progression (A.P):

According to given condition;

ii.

We are given that for the same Arithmetic Progression (A.P);

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

Therefore;

Hence;

We have found in (i) that;

**b.
**

From the given information, we can compile following data about Geometric Progression (G.P);

Expression for the sum to infinity of the Geometric Progression (G.P) when or ;

Expression for the sum of number of terms in the Geometric Progression (G.P) when is:

Hence;

Expression for the general term in the Geometric Progression (G.P) is:

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