IB Math HL – Sequences & Series

IB Math HL – Sequences & Series

Questions and Answers

1. IB Mathematics HL – Arithmetic Sequences and Series

How can we find the sum of the first 44 terms of the arithmetic sequence defined by the following formula?

a_{n}=8+4n for n \geq 0

Solution

IB Math HL – Arithmetic Sequences and Series

This is an arithmetic sequence with common difference d which can be found as follows

d=a_{n+1}-a_{n}=8+4(n+1)-(8+4n)=4

The first term is a=8, the common difference is d = 4, and n =44.

In order to find the sum of the first n terms, we are using the following formula for an arithmetic series:

S_{n}=\frac{n}{2}[2a+(n-1)d], where in our case a=8, d=4, n=44

Therefore, S_{44}=\frac{44}{2}[2(8)+(44-1)4]=\frac{44}{2}[16+172]=

=22(188)=4136

2. IB Mathematics HL – Geometric Sequence

A runner increases his running distance by 10% each week. If he

runs 4 kilometers the first week, find the total distance he ran after 9 weeks?

Solution

IB Mathematics HL – 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 10%.

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 1.10.

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

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

where   a=4, r=1.1, n=9

So, S_{8}=\frac{4((1.1)^8-1)}{1.1-1}=45.7 Km (3 s.f.)

3. IB Mathematics HL – Arithmetic Sequence

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

There are 60 of the marbles in the first row with three less in each row. How

many marbles are in the 16th row?

Solution

IB Mathematics HL – Arithmetic Sequences (Progressions)

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

marbles decreases by 3, the common difference is -3 and d = -3.

The first term is 60, common difference is d = -3, and n =16.

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

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

where a=60, d=-3, n=16

So,  u_{16}=60+(16-1)(-3)=60-45=15 marbles

4. IB Mathematics HL – Arithmetic Series

How can I find the sum of the first 15 terms for the sequence defined by the following formula?

 u_{n}=-2+8n

Solution

IB Mathematics HL – Arithmetic Series

This is an arithmetic sequence with common difference d = 8.

The first term is 6, common difference is d = 8, and n =15.

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=6, d=8, n=15

So,  S_{15}=\frac{15}{2}[2(6)+(15-1)8]= \frac{15}{2}[12+(14)8]=

 = \frac{15}{2}[124]=62(15)=930

Sequences and Series on the TI83/84 Graphing Calculator

 


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