IB Math SL -Sequences & Series

Questions and Answers

1. IB Math SL – Arithmetic Series

How can we find the sum of the first 24 terms of the sequence defined by the following formula?

$a_{n}=-4+5n$ for $n \geq 0$

Solution

IB Mathematics SL – Arithmetic Series

This is an arithmetic sequence with a common difference $d$ which can be found as following

$d=a_{n+1}-a_{n}=-4+5(n+1)-(-4+5n)=5$

The first term is $a=-4$ , the common difference is $d = 5$ , and $n =24$ .

In order to find the sum of the first n terms, you will use the following formula for an arithmetic series:

$S_{n}=\frac{n}{2}[2a+(n-1)d]$ , where in our case $a=-4, d=5, n=24$

Therefore, $S_{24}=\frac{24}{2}[2(-4)+(24-1)7]=\frac{24}{2}[-8+(23)5]=$

$=\frac{24}{2}[107]=12(107)=1284$

2. IB Mathematics SL – Geometric Sequence

A runner increases his running distance by 15% each week. If he

runs 6 kilometers the first week, find the total distance he ran after 11 weeks?

Solution

IB Mathematics 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 15%.

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 1.15.

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

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

where $a=6, r=1.15, n=11$

So, $S_{11}=\frac{6((1.15)^{11}-1)}{1.15-1}=146 Km (3 s.f.)$

3. IB Mathematics SL – Arithmetic Sequence

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

There are 74 of the marbles in the first row with four less in each row. How many marbles are in the 14th row?

Solution

IB Mathematics SL – Arithmetic Sequences (Progressions)

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

marbles decreases by 4, the common difference is d = -4.

The first term is 74, common difference is d = -4, and n =14.

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

$u_{n}=a+(n-1)d$ , where $a=74, d=-4, n=14$

So, $u_{14}=74+(14-1)(-4)=74-52=22$ marbles

4. IB Mathematics SL – Arithmetic Series

How can I find the sum of the first 16 terms for the sequence defined by the following formula?

$u_{n}=-2+7n$

Solution

IB Mathematics SL – Arithmetic Series

This is an arithmetic sequence with common difference d = 7.

The first term is 5, common difference is d = 7, and n =16.

To find the sum of the first n terms, you will use the following formula for the arithmetic series:

$S_{n}=\frac{n}{2}[2a+(n-1)d]$ , where $a=5, d=7, n=16$

So, $S_{16}=\frac{16}{2}[2(5)+(16-1)7]= \frac{16}{2}[10+(15)7]=$

$= \frac{16}{2}[115]=8(115)=920$