Questions and Answers

IB Math Studies Sequences

1. IB Mathematical Studies SL – Geometric Sequence

A runner increases his running distance by 5% each week. If he

runs 3 kilometers the first week, find the total distance he ran after 6 weeks?

Solution

IB Mathematical Studies SL – Geometric Sequences (Progressions)

A geometric sequence is determined by its initial term (first term) and its common ratio.

The general nth term of a geometric sequence with initial term $u_{1}$ and common ratio r is given by:

$u_{n}=u_{1}r^{n-1}$

This situation is represented by a geometric sequence because the quantities are increased by multiplying by 5%.

You are asked to find the total distance. This means you are looking for the sum of the geometric sequence. Because the distance ran increases every day above the previous day’s distance, the ratio will be 1.05.

To find the sum, you will use the following formula for a

geometric series:

$S_{n}=\frac{a(r^n-1)}{r-1}$

where $a=3, r=1.05, n=6$

So, $S_{6}=\frac{3((1.05)^6-1)}{1.05-1}=20.4 Km (3 s.f.)$

2. IB Mathematical studies– Arithmetic Sequences

An artist is creating a triangularly shaped sculpture made of marbles.

There are 24 of the marbles in the first row with one less in each row. How

many marbles are in the 5th row?

Solution

IB Mathematical Studies SL – Arithmetic Sequences (Progressions)

This situation is represented by an arithmetic sequence. Since the number of

marbles decreases by 1, the common difference is d = -1.

The first term is 24, common difference is d = -1, and n =5.

To find the nth term, you will use the following formula for a

arithmetic sequence:

$u_{n}=a+(n-1)d$

where $a=24, d=-1, n=5$

So, $u_{5}=24+(5-1)(-1)=24-4=20 marbles$

3. IB Mathematical Studies SL – Arithmetic Series

How can I find the sum of the first 6 terms for the sequence defined by the following formula?

$u_{n}=3+5n$

Solution

IB Mathematical Studies SL – Arithmetic Series

This is an arithmetic sequence with common difference d = 5.

The first term is 8, common difference is d = 5, and n =6.

To find the sum of the first n terms, you will use the following formula for a

arithmetic series:

$S_{n}=\frac{n}{2}[2a+(n-1)d]$

where $a=8, d=5, n=6$

So, $S_{6}=\frac{6}{2}[2(8)+(6-1)5]= \frac{6}{2}[16+25]=$

$= \frac{6}{2}[41]=3(41)=123$