Sequences Exercises

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The first thing we'll do for the sequence is make a table to help us see the relationship between n and a_n.

The second number in each row is 2 greater than the first number. The formula for the general term is

a_n = n + 2.

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The first thing we'll do for the sequence is make a table to help us see the relationship between n and a_n.

The n^th term is a_n = 4n.

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The first thing we'll do for the sequence is make a table to help us see the relationship between n and a_n.

This sequence is obtained by adding 1 to each term of the previous sequence:

a_n = 4n + 1.

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The first thing we'll do for the sequence is make a table to help us see the relationship between n and a_n.

These look almost like squares, just slightly out of place.

The formula for the general term is a_n = n² – 1.

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The first thing we'll do for the sequence is make a table to help us see the relationship between n and a_n.

The general term is a_n = 3n.

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The first thing we'll do for the sequence is make a table to help us see the relationship between n and a_n.

These terms are very close to the terms in the previous sequence.

The general term is a_n = 3n – 1.

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The first thing we'll do for the sequence is make a table to help us see the relationship between n and a_n.

We don't need to make a table for this one. The n^th term is -n. As a formula,

a_n = -n.

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The first thing we'll do for the sequence is make a table to help us see the relationship between n and a_n.

Each term is 1 less than a multiple of 10. Which multiple of 10 depends on n.

The formula for the general term is a_n = 10n – 1.

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The first thing we'll do for the sequence is make a table to help us see the relationship between n and a_n.

If you look at this as similar to the previous sequence, each term is some amount less than a multiple of 10. Which multiple of 10, and the amount less, both depend on n.

The general term is

a_n = 10n – n.

If you look at this sequence from scratch, you're more likely to see

Now the general term is a_n = 9n.

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The first thing we'll do for the sequence is make a table to help us see the relationship between n and a_n.

These numbers are all even, so we'll guess that they're 2 multiplied by something.

The general term is a_n = 2n².

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The numerators are odd numbers:

This means the numerator is (2n – 1). The denominators are powers of 2:

This means the denominator is 2ⁿ.

The formula for the general term is

.

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The numerators are cubes:

The numerator is given by n³.

The denominators are multiples of 3:

The denominator is given by 3n.

The formula for the general term is

We could also simplify this to

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The numerators are even numbers and the denominators are odd numbers. We can rewrite the sequence as

The formula for the general term is

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The numerators are even numbers, given by 2n. Each denominator is 3 greater than its corresponding numerator:

The formula for the general term is

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The numerators are even numbers again, given by 2n. The denominators are multiples of 3, but starting at 6 instead of 3. Let's make a table to see what's going on.

The denominator is given by 3(n + 1). The formula for the general term is

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If we ignore the signs we see the sequence of even numbers 2n. In order to have the signs alternate with the first term positive, we need to multiply by (-1)^{n + 1}.

The general term is

a_n = (-1)^{n + 1}2n

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If we ignore the signs we see the sequence is made of multiples of 3. In order to have the signs alternate with the first term negative, we need to multiply by (-1)ⁿ.

The general term is

a_n = (-1)ⁿ3n

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If we ignore the signs we see the odd numbers (2n – 1). To have alternating signs with the first term negative, we multiply by (-1)ⁿ.

The general terms is

a_n = (-1)ⁿ(2n – 1).

Make sure you have proper parentheses in this expression, because

(-1)ⁿ2n – 1

means something different!