Sequences Exercises

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Look at the ratios between successive terms. If the ratios are all the same, the sequence is geometric. If not, it isn't.

This IS a geometric sequence.

Answer

Look at the ratios between successive terms. If the ratios are all the same, the sequence is geometric. If not, it isn't.

We don't have to go any further. The ratios aren't all the same, so this is NOT a geometric sequence.

Answer

Look at the ratios between successive terms. If the ratios are all the same, the sequence is geometric. If not, it isn't.

The ratios are all the same, so this IS a geometric sequence.

Answer

Look at the ratios between successive terms. If the ratios are all the same, the sequence is geometric. If not, it isn't.

This IS a geometric sequence.

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Look at the ratios between successive terms. If the ratios are all the same, the sequence is geometric. If not, it isn't.

The ratios aren't the same, so this is NOT a geometric sequence.

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Any ratio between successive terms is either

or

Since the ratio is always -1, this IS a geometric sequence.

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Since the ratio between successive terms is always -7, this IS a geometric sequence.

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The ratios between successive terms are not always the same. This is NOT a geometric sequence.

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To get from one term to the next, multiply by 3. The first four terms are

7, 21, 63, 189

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To get from one term to the next, multiply by  (equivalently, divide by 5). The first four terms are

100, 20, 4, 0.8

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To get from one term to the next, multiply by 0.5 (equivalently, divide by 2). The first four terms are

2, 1, 0.5, 0.25

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To get from one term to the next, multiply by -4:

8, -32, 128, -512

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Since we're multiplying by 1, all the terms are the same:

3, 3, 3, 3

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To get from one term to the next we multiply by 5. In order to see the pattern, we recommend NOT multiplying the numbers together, but keeping the expression for each term in factored form.

a) a₂ = a₁(5) = 4 × 5

b) a₃ = a₂ × 5= 4 × 5 × 5= 4 × 5²

c) a₄ = a₃ × 5= 4 × 5² × 5= 4 × 5³

d) We haveWe conclude thata_n = 4 × 5^{n – 1}.

Answer

To get from one term to the next we multiply by . In order to see the pattern, we recommend keeping terms in factored form.

a) 

b)

c) 

d) We have We must have

Answer

We don't have numbers for a₁ and r, but other than that, nothing has changed.

a) a₂ = a₁r

b) a₃ = a₂r= a₁rr= a₁r²

c) a₄ = a₃r= a₁r²r= a₁r³

d) Look at the terms we have so far:This means the general formula for the n^th term of a geometric sequence isa_n = a₁r^{n – 1}.

Answer

We have a = 8, r = 0.2 and we want to find a₅. We plug the numbers into the appropriate places:

a_n = ar^{n – 1}

a₅ = (8)(0.2)⁴

= 0.0128.

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We plug the numbers into the appropriate places in the formula:

a_n = ar^{n – 1}

a₇ = (2)(-5)⁶

= 31,250.

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Plug in the numbers:

a_n = ar^{n – 1}

a₂₀ = (0.125)(2)²⁰

= 131,072.

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The common ratio is

Plugging this and a₇ = 640 into the formula, we get

a_n = ar^{n – 1}

a₇ = a(-2)⁶

640 = a(64)

We conclude that

Answer

The common ratio is

Plugging things into the formula, we get

a_n = ar^{n – 1}

a₄ = a(0.2)³

1.2 = a(.008)

We conclude that

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The common ratio is

Using the nice formula, we get

a_n = ar^{n – 1}

a₄ = a(-0.25)³

-3.125 = a(-0.015625)

We conclude that

Answer

Find the ratio between the given terms:

Since there are 2 multiplications from a₁₁ to a₁₃, take the square root to get r:

r = (0.0625)^1/2 = 0.25

Then use the formula to find a.

a_n = ar^{n – 1}

a₁₁ = ar¹⁰

1 = a(0.25)¹⁰

We conclude that

Answer

Find the ratio between the given terms:

Since there are 3 multiplications between these terms, take the third root to get r:

r = (0.512)^1/3 = 0.8

Plug things into the geometric sequence formula.

a_n = ar^{n – 1}

a₄ = ar³

51.2 = a(0.8)³

We conclude that