## Savings Schemes

Many saving schemes require a regular deposit to be made in each set time period – typically week, month or year. A very good example is saving for a pension. Money is deducted from your wages and invested. It is hoped that over a working life enough will be saved to provide a good income in retirement.

If the rate of interest is fixed, we can reduce the problem to consideration of a geometric series. Suppose for example that £1,000 is invested at the start of each year for a period of ten years. Money is paid at the end of each year on the money in the account at the start of the year.

At the start of the 1st year £1,000 is invested, gathering interest for the entire ten years. At the end of the ten years this £1,000 will have become

At the start of the 2nd year £1,000 is invested, gathering interest for the last nine years. At the end of the ten years this £1,000 will have become

At the start of the 3rd year £1,000 is invested, gathering interest for the last eight years. At the end of the ten years this £1,000 will have become

At the start of the 4th year £1,000 is invested, gathering interest for the last seven years. At the end of the ten years this £1,000 will have become

At the start of the 5th year £1,000 is invested, gathering interest for the last six years. At the end of the ten years this £1,000 will have become

At the start of the 6th year £1,000 is invested, gathering interest for the last five years. At the end of the ten years this £1,000 will have become

At the start of the 7th year £1,000 is invested, gathering interest for the last four years. At the end of the ten years this £1,000 will have become

At the start of the 8th year £1,000 is invested, gathering interest for the last three years. At the end of the ten years this £1,000 will have become

At the start of the 9th year £1,000 is invested, gathering interest for the last two years. At the end of the ten years this £1,000 will have become

At the start of the 10th year £1,000 is invested, gathering interest for the last year. At the end of the ten years this £1,000 will have become

The total amount has grown over the period of ten years to the sum of all the above expressions. It is convenient to write the sum in reverse order:

Obviously the terms form a geometric sequence with first termand common ratioThere are ten terms so, using the formula for the sum of a geometric series,

we have