Arithmetic Sequences and Series

Arithmetic Sequences and Series

Math AISL

​General Sequences

​A sequence of numbers is a list of numbers (of finite or infinite length) arranged in order that obeys a certain rule.

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​Each number in the sequence is called a term. The nth term, where n is a positive integer, can be represented by the notation u_n.

Open Ended

What is the general rule for this sequence? 

2, 4, 6, 8, 10, ...

Open Ended

What is the general rule for this sequence?

1, 1/2, 1/3, 1/4, ...

Open Ended

Write down the next three terms in this sequence: 100, 75, 50, 25, ...

Open Ended

Write down the first three terms of this sequence:  $$u_n=n+1$$  

(n is a positive integer,  $$Z^+$$  )

Open Ended

Consider the sequence $$u_n=3+\left[4\left(n-1\right)\right]$$  . Find the value of n for which $$u_n=111$$  .

​Arithmetic Sequences

​A sequence in which the difference between each term and its previous one remains constant.

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​The constant difference is called the common difference.

Poll

Which of these sequences are arithmetic sequences? Choose all correct answers.

• 2, 4, 6, 8, 10, ...

• 1, 10, 100, 1000, ...

• -1, -0.5, 0, 0.5, ...

• $$u_n=5n+2$$  

• $$r_n=n^2$$  

​The general term of an arithmetic sequence

​An arithmetic sequence with first term u₁ and common difference d can be generated as:

​u₁

​u₂ = u₁ + d

​u₃ = u₁ + d + d = u₁ + 2d

​u₄ = u₁ + d + d + d = u₁ + 3d

​u₅ = u₁ + d + d + d + d = u₁ + 4d

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​Following the pattern, u_n = u₁ + (n-1) d, where n is a positive integer.

Open Ended

Consider this finite arithmetic sequence: -3, 5, ... , 1189.

Write down the common difference.

Open Ended

Remember that the general term of an arithmetic sequence is u_n = u₁ + (n-1) d.

Find the number of terms in the sequence: -3, 5, ... , 1189

​​GDC use for the previous question

​Find the number of terms in the sequence: -3, 5, ... , 1189.

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​Step 1: Go to Menu and select EQUATION.

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​​Step 2: Press (F3) SOLVE.

​Step 3: Type in the equation: [(-3) + ((n-1)*8)] = 1189

​Arithmetic Series

​Example

Open Ended

Find the sum of the arithmetic series (-10) + (-6) + (-2) + ... + 90.

Use the formula:

$$S_n=\frac{n}{2}\left[2u_1+\left(n-1\right)d\right]$$

Open Ended

Find the least number of terms that must be added to the series (-10) + (-6) + (-2) + ... to obtain a sum greater than 100.

(GDC use is recommended.)

​GDC use for the previous question

​Find the least number of terms that must be added to the series (-10) + (-6) + (-2) + ... to obtain a sum greater than 100.

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Given: d = 4 and sum > 100​ Step 2: Set (F5) your GDC as follows:

Find: n (number of terms)​

Step 1: Go to menu and select TABLE​.

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​Continuation

​​ Step 3: Press F6 (table function)

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Adding 10 terms gives a sum of 80.

​Adding 11 terms gives a sum of 110, so n = 11.

​Definition of Terms

​1. Capital - the money that you put in a savings institution or a bank

​2. Interest

​a. If you put money in a bank, the bank will pay you interest.

​b. If you borrow money from a bank or savings, institution, then you have to pay them interest.

​3. Simple Interest - simplest way to calculate interest

​Example 1:

​You want to borrow USD 1000 for three years at a rate of 4% per year.

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​Interest for 1 year: 0.04 x 1000 = USD 40

​Interest for 2 years: (0.04 x 1000) x 2 = USD 80

​Interest for 3 years: (0.04 x 1000) x 3 = USD 120

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​Simple Interest Formula: I = Crn

​C: capital

​r: interest rate

​n: number of interest periods

​I: interest

​Example 2:

​An amount of UK£5000 in invested at a simple interest rate of 3% per annum for a period of 8 years.

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​a. Calculate the interest received after 8 years.

​Given: C = 5000 ; r = 0.03 ; n = 8

​Using the formula, I = Crn

​ I = 5000 x 0.03 x 8 --> I = UK£1200 (interest received in 8 years)

​b. Find the total amount in the account after the 8 years.

​To find the total amount, add the interest to the capital.

​UK£5000 + UK£1200 = UK£6200

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Example 3:

​You invested USD4000 in a bank account at a simple interest rate of 7.3% per year. Find the number of years it takes for your money to double.

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Calculate the simple interest on a loan of USD10000 at a rate of 8.5% pa (per annum) over 18 months.

Note: 18 months is 1.5 years

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Calculate the simple interest on a loan of USD6500 at a rate of 7% pa (per annum) over 3 years and 5 months.