# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2012 | January | Q#9

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**Question**

A company offers two salary schemes for a 10-year period, Year 1 to Year 10 inclusive.

Scheme 1: Salary in Year 1 is £P.

Salary increases by £(2T) each year, forming an arithmetic sequence.

Scheme 2: Salary in Year 1 is £(P+1800).

Salary increases by £T each year, forming an arithmetic sequence.

**a. **Show that the total earned under Salary Scheme 1 for the 10-year period is £(10P+90T)

For the 10-year period, the total earned is the same for both salary schemes.

**b. **Find the value of T.

For this value of T, the salary in Year 10 under Salary Scheme 2 is £29 850

**c. **Find the value of P.

**Solution**

**a.
**

We are given that the salaries of all 10 years form an arithmetic sequence.

From the given information we can collect following data about the said arithmetic sequence.

Total salaries are over a 10-year period, from Year 1 to Year 10 inclusive.

Salary for year 1 is £P.

Salary increases by £(2T) each year.

We are required to find the total salary earned over a period of 10 years.

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

Therefore;

Hence, total salary earned under scheme 1 over a period of 10 years is £ (10P+90T).

**b.
**

We are given that for the 10-year period, the total earned is the same for both salary schemes.

We have already found total salary earned under scheme 1 over a period of 10 years is £ (10P+90T).

Since for the 10-year period, the total earned is the same for both salary schemes, we need to find the total salary over 10-year period for scheme 2 as well.

We are given that the salaries of all 10 years form an arithmetic sequence.

From the given information we can collect following data about the said arithmetic sequence.

Total salaries are over a 10-year period, from Year 1 to Year 10 inclusive.

Salary for year 1 is £(P+1800).

Salary increases by £T each year.

We are required to find the total salary earned over a period of 10 years.

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

Therefore;

Hence, total salary earned under scheme 1 over a period of 10 years is £ (10P+45T+18000).

We are given that for the 10-year period, the total earned is the same for both salary schemes.

Therefore;

**c.
**

For scheme 2 we have found in (b) that;

We are given that salary of Year 10 is £29850.

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

Therefore;

As demonstrated in (b), T=400, hence;

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