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Math 4: Geometric and Algebraic sequences

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Content

• Arithmetic sequences

• Example

• Geometric sequences

• Example

• Practice exercises

• Answers

Arithmetic sequences

A number sequence (or progression) can be defined as a set of ordered numbers. Each of these numbers are called terms of the sequence: a1 is the first term, a2 is the second term, a3 is the third term, and so on, establishing that an is the n-th term. The sequences can be finite or infinite depending on the number they have. They can also be increasing if each term is greater than its previous one (an ≤ an + 1) or decreasing if each term is less than its previous one (an ≥ an + 1).

It is called an arithmetic sequence when each term is the addition or subtraction of the previous term plus or minus a constant number, which we call the difference and denote by the letter d.

The general formula of the progression allows us to indicate the value of any term in the sequence without having to write the previous terms. Similarly, we can calculate the sum of n number of consecutive terms.

The formula to obtain the value of any term of the sequence: an = a1 + d(n - 1)

Formula to obtain the value of the difference when having two terms: d = an+1 - an

Knowing the first term and the n-th term of the sequence, we can calculate the sum of the first n terms with the formula: Sn = n(a1 + an)/2

You can also calculate the sum of the first n terms from the first a1, and the difference with the formula: Sn = n(a1)[(d)(n)(n - 1)/2]

Example

  • What is the second term of the following arithmetic sequence?

5, a2, 21, 29, ...

d = 8
a1 = 5
a2 = ?

an = a1 + d(n - 1)
a2 = 5 + 8(2 - 1)
a2 = 5 + 8
a2 = 13

  • Calculate the term a1 of the arithmetic sequences from the given data.

1. a4 = 11,
a5 = 14

2. a4 = 23
d = 4

3. a4 = 28
a6 = 34


an = a1 + d(n - 1)
an - d(n - 1) = a1
a1 = an - d(n - 1)

1. a1 = an - d(n - 1)
a1 = 11 - 3(4 - 1)
a1 = 11 - 3(3)
a1 = 11 - 9
a1 = 2

2. a1 = an - d(n - 1)
a1 = 23 - 4(4 - 1)
a1 = 23 - 4(3)
a1 = 23 - 12
a1 = 11

3. a6 - a4 = 2d
34 - 28 = 6
d = 6/2 = 3

a1 = an - d(n - 1)
a1 = 34 - 3(6 - 1)
a1 =34 - 3(5)
a1 =34 - 15
a1 =19

  • Calculate how many odd numbers there are between 20 and 50 and calculate their sum.

Odd numbers begin with 21, which means a1 = 21, the following number (a2) is 23, and the difference between this numbers is d = 2.
The last odd number before 50 is 49, taking that an = 49 and a1 = 21, we can obtain n, which represents the number of odd numbers between 20 and 50.

an = a1 + d(n - 1)
an - a1 = d(n - 1)
[(an - a1) /d] + 1 = n

n = [(49 - 21) /2] + 1
n = [28/2] + 1
n = 14 + 1
n = 15

Sn = n(a1 + an)/2
S15 = 15(21 + 49)/2
S15 = 15(70)/2
S15 = 1050/2
S15 = 525

Geometric sequences

A geometric sequence, on the other hand, is a sequence in which each term an is obtained by multiplying the previous term an − 1 by a number called a ratio, named with the letter r and which can be a rational number (multiplying by a rational number It is equivalent to dividing by another number, for example multiplying by 0.5 is equivalent to dividing by 2).

The formula to obtain the value of any term of the sequence: an = (a1)(rn-1)

Formula to obtain the value of the ratio when having two terms: r = an+1 / an

Knowing the first term and the ratio, we can calculate the sum of the first n terms with the formula: Sn = a1[(rn - 1)/(r - 1)]

You can also calculate the sum of the first n terms with the formula: Sn = [(an)(r - a1)]/r - 1

Example

  • Calculate the fifth term of the geometric progression

3/5, 6/25, 12/125, ...

r = an+1 / an
r = 6/25 / 3/5
r = 6/25 / 3/5
r = 0.4

an = (a1)(rn-1)
a5 = (3/5)(2/55-1)
a5 = (3/5)(2/54)
a5 = (3/5)(16/625)
a5 = (3/5)(16/625)
a5 = 48/3125
a5 = 0.01536

  • Calculate the sum of the first 5 terms of the following geometric sequence:

2, 6, 18, ...

r = an+1 / an
r = 6 / 2
r = 3

an = (a1)(rn-1)
a5 = (2)(35-1)
a5 = (2)(34)
a5 = (2)(34)
a5 = (2)(81)
a5 = 162

Sn = a1[(rn - 1)/(r - 1)]
S5 = 2[(35 - 1)/(3 - 1)]
S5 = 2[(35 - 1)/(2)]
S5 = 2[(243 - 1)/(2)]
S5 = 2[242/2]
S5 = 2[121]
S5 = 242

Practice exercises

  • Sum the first 12 terms of the following geometric sequence

5, 25, 125, ...


  • Find an arithmetic sequence which first term is 3 and which first three terms add up to 12.

Answers

  • Sum the first 12 terms of the following geometric sequence

5, 25, 125, ...

r = a3 / a2
r = 125 / 25
r = 5

an = (a1)(rn-1)
a12 = (5)(512-1)
a12 = (5)(511)
a12 = (5)(48,828,125)
a12 = 244,140,625

Sn = a1[(rn - 1)/(r - 1)]
S12 = 5[(512 - 1)/(5 - 1)]
S12 = 5[(244,140,625 - 1)/(5 - 1)]
S12 = 5[244,140,624/4]
S12 = 5[61,035,156.25]
S12 = 305,175,781.3

  • Find an arithmetic sequence which first term is 3 and which first three terms add up to 12.

a1 = 3
S3 = 12
d = ?
a3 = ?

Sn = n(a1 + a3)/2
12 = 3(3 + a3)/2
(12)(2) = 3(3 + a3)
24/3 = 3 + a3
8 - 3 = a3
a3 = 5

d= (a3 - a1)/2
d = (5 - 3)/2
d = 2/2
d = 1

© 2021 Daniela Alejandra Rodríguez Cerda

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