1. 3. Convergent Series & Compound Interest
Practice Questions
Ex 3.4: Q 6, 7 p99
Ex 3.3: Q 15, 16, 17 p90

2. Convergent Geometric Series
If a geometric series has a common ratio between -1 and 1, then the terms get smaller and smaller as n increases.
Formula for the sum of a geometric sequence is:
as n gets larger (approaches ∞) then r n gets smaller (approaches 0)
if we sum the sequence to ∞, then we can substitute 0 for r n

3. Sum to Infinity Examples
Find the sum to infinity of the series:
Write as a rational number.
3. Use an infinite series to express the recurring decimal as a rational number.

4. Sum to Infinity Examples
A ball is dropped from a height of 10 metres. On each bounce the ball bounces to three quarters of the height of the previous bounce. Find the distance travelled by the ball before it comes to rest (assume it does not move sideways).
The sum of the first three terms of a geometric series is 148, and the sum to infinity is 256.
Find the first term and the common ration of the series.

5. Compound Interest
Compound interest is interest that is earned on the original investment plus any interest previously eared.
Eg. $1000 invested at 10% p.a. grows as follows: u1= 1000
first year = $1000(1.10) = $1100
second year = $1100(1.10) = $1000(1.10)2 = $ u1(r n-1)
third year = $1210(1.10) = $1000(1.10)3 = $1331
nth year =
Compound interest can be calculated using the following variation of the geometric sequence formula:
Where:
U0 = initial amount invested
yn = amount after n years
r = interest rate as a decimal + 1
REMEMBER: term 1 = year 0

6. Compound Interest Examples
$5000 is invested for 4 years at 7% p.a. compound interest. What will the investment amount to at the end of this period?
7. How much should I invest now if I want $10000 in 4 years time, if I am able to invest at 8.5% p.a. Compounded annually?

7. Compound Interest Examples
A bank account offers 9% interest compounded annually. If $750 is invested in this account, find the total amount of interest earned at the end of the 12th year.
9. The population of a city is people. The population is increasing at a rate of 1.3% per year. Assuming the population continues to increase at this rate, how long will it take for the population of the city to reach people?

8. Different Compounding Periods
If $1000 is invested at the rate of 8% p.a. Compounded monthly, how much will be in the account after the end of 5 years?
When the compounding period is not yearly:
c = the number of compounding periods per year
Interest rate per compounding period
= interest per year =
# of compound periods/year
# of compound periods
= # years x # compound periods/year
= n x c