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Arithmetic and Geometric Sequences (page 3 of 5)

Sections: Terminology and notation, Basic examples, Arithmetic and geometric sequences, Arithmetic series, Finite and infinite geometric series

The two simplest sequences to work with are arithmetic and geometric sequences. An arithmetic sequence goes from one term to the next by always adding (or subtracting) the same value. For instance, 2, 5, 8, 11, 14,... and 7, 3, –1, –5,... are arithmetic, since you add 3 and subtract 4, respectively, at each step. A geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value. So 1, 2, 4, 8, 16,... and 81, 27, 9, 1, 1/3,... are geometric, since you multiply by 2 and divide by 3, respectively, at each step.

The number added (or subtracted) at each stage of an arithmetic sequence is called the "common difference" d, because if you subtract (find the difference of) successive terms, you'll always get this common value. The number multiplied (or divided) at each stage of a geometric sequence is called the "common ratio" r, because if you divide (find the ratio of) successive terms, you'll always get this common value. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

• Find the common difference and the next term of the following sequence:

3, 11, 19, 27, 35,...

To find the common difference, I have to subtract a pair of terms. It doesn't matter which pair I pick, as long as they're right next to each other:

11 – 3 = 8
19 – 11 = 8
27 – 19 = 8
35 – 27 = 8

The difference is always 8, so d = 8. Then the next term is 35 + 8 = 43.

• Find the common ratio and the seventh term of the following sequence:

2/9, 2/3, 2, 6, 18,...

To find the common ratio, I have to divide a pair of terms. It doesn't matter which pair I pick, as long as they're right next to each other:

The ratio is always 3, so r = 3. Then the sixth term is (18)(3) = 54 and the seventh term is (54)(3) = 162.

Since arithmetic and geometric sequences are so nice and regular, they have formulas.

For arithmetic sequences, the common difference is d, and the first term a₁ is often referred to simply as "a". Since you get the next term by adding the common difference, the value of a₂ is just a + d. The third term is a₃ = (a + d) + d = a + 2d. The fourth term is a₄ = (a + 2d) + d = a + 3d. Following this pattern, the n-th term a_n will have the form a_n = a + (n – 1)d.

For geometric sequences, the common ratio is r, and the first term a₁ is often referred to simply as "a". Since you get the next term by multiplying by the common ratio, the value of a₂ is just ar. The third term is a₃ = r(ar) = ar². The fourth term is a₄ = r(ar²) = ar³. Following this pattern, the n-th term a_n will have the form a_n = ar^{(n – 1)}.

• Find the tenth term and the n-term of the following sequence:

1/2, 1, 2, 4, 8,...

The differences don't match: 2 – 1 = 1, but 4 – 2 = 2. So this isn't an arithmetic sequence. On the other hand, the ratios are the same: 2 ÷ 1 = 2, 4 ÷ 2 = 2, 8 ÷ 4 = 2. So this is a geometric sequence with common ratio r = 2 and a = 1/2. To find the tenth and n-th terms, I can just plug into the formula:

a_n = (1/2)2^n–1
a₁₀ = (1/2)2^10–1 = (1/2)2⁹ = (1/2)(512) = 256

• Find the n-th term and the first three terms of the arithmetic sequence having a₆ = 5 and d = 3/2.

The n-th term is of the form a_n = a + (n – 1)d. In this case, that gives me a₆ = a + (6 – 1)(3/2) = 5. Solving, I get a = –5/2. Then:

a₁ = –5/2, a₂ = –5/2 + 3/2 = –1, a₃ = –1 + 3/2 = 1/2,
and a_n = –5/2 + (n – 1)(3/2)

• Find the n-th term and the first three terms of the arithmetic sequence having a₄ = 93 and a₈ = 65.

Since a₄ and a₈ are four places apart, then I know from the definition of an arithmetic sequence that a₈ = a₄ + 4d. Then:

65 = 93 + 4d
–28 = 4d
–7 = d

Also, a₄ = a + (4 – 1)d, so:

93 = a + 3(–7)
93 + 21 = a
114 = a

Then a₁ = 114, a₂ = 114 – 7 = 107, a₃ = 107 – 7 = 100, and a_n = 114 + (n – 1)(–7)

• Find the n-th and 26th terms of the geometric sequence with a₅ = 5/4 and a₁₂ = 160.

These two terms are 12 – 5 = 7 places apart, so, from the definition of a geometric sequence, I know that a₁₂ = (a₅)(r⁷). Then:

160 = (5/4)(r⁷)
128 = r⁷
2 = r

Since a₅ = ar⁴, then:

5/4 = a(2⁴) = 16a
5/64 = a

Then a_n = (5/64)2^{(n – 1)} and a₂₆ = (5/64)(2²⁵) = 2 621 440

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