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Sequences and Series: Basic Examples (page 2 of 5)

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

• Let A_n = {1, 3, 5, 7, 9}. What is the value of a₃?
  Find the value of   

The index of a₃ is n = 3, so they're asking me for the third term, which is "5". The "value" they're asking for is the total, the sum, of all the terms a_n from a₁ to a₅; in other words:

a₁ + a₂ + a₃ + a₄ + a₅ = 1 + 3 + 5 + 7 + 9 = 25

value of a₃: 5
value of sum: 25

• Expand the following series and find the sum:

To find each term, I'll plug the value of n into the formula. In this case, I'll be starting with
n = 0 and ending with n = 4.

2(0) + 2(1) + 2(2) + 2(3) + 2(4) = 0 + 2 + 4 + 6 + 8 = 20

• List the first four terms of the sequence {a_n} = {n²}, starting with n = 1.

I'll just plug n into the formula, and simplify:

{a₁, a₂, a₃, a₄} = {1², 2², 3², 4²} = {1, 4, 9, 16}

• List the first four terms of the following sequence, beginning with n = 0:

Many sequences and series contain factorials, and this is one of them. I'll evaluate in the usual way:

So the terms are: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Notice how, in that last example above, raising the –1 to the power n made the signs alternate. This alternation of signs crops up a lot, especially in calculus, so try to keep this trick in mind.

• Find the sum of the first six terms of A_n, where a_n = 2a_n–1 + a_n–2, a₁ = 1, and a₂ = 1.

This looks much worse than it really is. They gave me the values of the first two terms, and then gave me a formula that says that each term (after the first two) is a sum formed from the previous two terms. Plugging into the formula, I get:

a₃ = 2a_3–1 + a_3–2 = 2a₂ + a₁ = 2(1) + (1) = 2 + 1 = 3
a₄ = 2a_4–1 + a_4–2 = 2a₃ + a₂ = 2(3) + (1) = 6 + 1 = 7
a₅ = 2a_5–1 + a_5–2 = 2a₄ + a₃ = 2(7) + (3) = 14 + 3 = 17
a₆ = 2a_6–1 + a_6–2 = 2a₅ + a₄ = 2(17) + (7) = 34 + 7 = 41

Then the sum is:

1 + 1 + 3 + 7 + 17 + 41 = 70

• Write the following series using summation notation, beginning with n = 1:

2 – 4 + 6 – 8 + 10

The first thing I have to do is figure out a relationship between n and the terms in the summation. This series is pretty easy, though: each term a_n is twice n, so there is clearly a "2n" in the formula. I also have the alternating sign. If I use (–1)ⁿ, I'll get –2, 4, –6, 8, –10, which is backwards (on the signs) from what I want. But I can switch the signs by throwing in one more factor of –1:

(–1)(–1)ⁿ = (–1)¹(–1)ⁿ = (–1)ⁿ⁺¹

So the formula for the n-th term is a_n = (–1)ⁿ⁺¹(2n). Since n starts at 1 and there are five terms, then the summation is:

• Write the following using summation notation:

The only thing that changes from one term to the next is one of the numbers in the denominator. (If I "simplify" these fractions, I'll lose this information. Any time the terms look oddly lumpy, I do not simplify, because that odd lumpiness almost certainly contains a hint of the pattern I need to find.) The changing numbers, as a list, are 6, 7, and 8. This looks like counting, but starting with 6 instead of 1. Without any information to the contrary, I'll assume that this is the pattern.

But I need to relate these "counting" values to the counter, the index, n. For n = 1, the number is 6, or n + 5. For n = 2, the number is 7, which is also n + 5. Checking the pattern for n = 3, 3 + 5 = 8, which is the third number. Then the terms seems to be in the following pattern:

But how many terms are in the summation? The ellipsis (the "..." or "dot, dot, dot" in the middle) means that terms were omitted. However, now that I have the general pattern for the series terms, I can solve for the counter (the value of n) in the last term:

31 = n + 5
31 – 5 = n + 5 – 5
n = 26

This tells me that there are 26 terms in this summation, so the series, in summation notation, is:

If the fractions (above) had been simplified and reduced, it would have been a lot harder to figure out a pattern. Unless the sequence is very simple or is presented in a very straightforward manner, it is possible that you don't be able to find a pattern, or might find a "wrong" pattern. Don't let this bother you terribly much: the "right" pattern is just the one that the author had in mind when he wrote the problem. Your pattern would be "wrong" only in that it is unexpected. But if you can present your work sensibly and mathematically, you should be able to talk your way into at least partial credit.

Fortunately, once you've learned the basic notation and terminology, you should quickly move on to the two common sequence types....

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