Sequences and series (SL) Flashcards

1
Q

What is an arithmetic sequence?

A

A list of numbers that change by adding or subtracting the same amount each time.

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2
Q

What is a “common difference, d”?

A

It’s the amount of increase (or decrease when negative) between terms of an arithmetic sequence.

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3
Q

How can you find d when given two consecutive terms (like u3 and u4) of an arithmetic series?

A

You subtract the terms. Always the later term minus the earlier one.

u4 - u3 = d

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4
Q

How can you find d when given two non-consecutive terms (like u4 and u8) of an arithmetic series?

A

You set up a system of equations and solve it (with linSolve).

  1. Write an equation for each un you know.
  2. Plug these equations into linSolve.
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5
Q

How can you find un if you already know u1 and d of an arithmetic series?

A

You plug it into the formula from your booklet.

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6
Q

What’s the difference between un and n?

A
  • n* tells you the placement of the term. Like, 1st, 2nd, 37th, etc.
  • un* is the actual value of the term.
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7
Q

What’s the difference between un and Sn?

A
  • un* is the value of only the nth term
  • Sn* is what you get if you add together all of the n terms.
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8
Q

What are context clues that you should be finding Sn?

A

You see these words in the question:

  • calculate/find the total
  • the sum of the first n terms…
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9
Q

Your formula booklet has two equations for Sn of an arithmetic sequence.

Which one should you use?

A

It all comes down to whether you know un or d.

  • Use the first one if you only know u1 and d.
  • Use the second one if you know both the 1st and last terms (u1 and un).
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10
Q

What is a geometric sequence?

A

A list of numbers that change by multiplying (or dividing) the same amount each time.

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11
Q

What is a “common ratio, r”?

A

It’s the multiple of increase (or decrease when a fraction less than 1) between terms of a geometric sequence.

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12
Q

How can you find d when given two consecutive terms (like u3 and u4) of an arithmetic series?

A

You divide the terms. Always the later term divided by the earlier one.

r = u4 ÷ u3

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13
Q

How can you find r when given two non-consecutive terms (like u4 and u8) of an geometric series?

A

You set up a system of equations and solve it (with nSolve).

  1. Write an equation for each un you know.
  2. Solve one equation for u1 by hand. (It will still have r in it).
  3. Substitute this equation into the other. (Now there are only rs.)
  4. Solve this equation with nSolve.
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14
Q

How can you find un if you already know u1 and r of a geometric series?

A

You plug it into the formula from your booklet.

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15
Q

Your formula booklet has two equations for Sn of a geometric sequence.

Which one should you use?

A

It literally makes no difference. Some people argue that one is better than the other, depending on whether r is greater or less than 1. But this is nonsense. Use either one; it doesn’t matter at all.

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16
Q

What type of sequence (arithmetic or geometric) is it if you are asked about simple interest?

A

Arithmetic!

17
Q

If the question talks about “percent increase” or “percent decrease”, what type of sequence is it?

A

Geometric!

r = 1 + “percent increase”

r = 1 - “percent decrease”

18
Q

How can you tell when a series exceeds some value?

A

You set up an equation for when Sn equals that value.

Nsolve to find n.

If n is a decimal number, check Sn for the integer values above and below the decimal number.

19
Q

What does “converge” mean for a geometric sequence?

A

It means the terms are getting smaller and smaller and eventually are so small, they’re simply 0.

20
Q

What has to be true about r for the infinite sum of a geometric series to exist?

A

r needs to be less than 1 in absolute value so that the terms are getting closer and closer to zero.

r | < 1