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 10th term.

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

From the given data, we can write expression for 10th 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 11th and 15th 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 11th term of Arithmetic Progression (A.P):

Similarly, from the given data, we can write expression for 11th 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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