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How do you find the nth term of the sequence #1/2, 1/4, 1/8, 1/16, ...#?

Answer 1

Emily Ammerman

#a_n=1/2^n#

We have #1/2,1/2^2,1/2^3, cdots, 1/2^n# so

#a_n = 1/2^n#

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Answer 2

Evelyn Agard

The nth term of the sequence (1/2, 1/4, 1/8, 1/16, \ldots) can be found using the formula for a geometric sequence:

[a_n = a_1 \cdot r^{(n-1)}]

where:

(a_n) is the nth term of the sequence,
(a_1) is the first term of the sequence,
(r) is the common ratio, which is the number each term is multiplied by to get the next term, and
(n) is the term number.

In this sequence, the first term (a_1) is (1/2), and the common ratio (r) is (1/2) because each term is half of the previous one.

Plugging these values into the formula, we get:

[a_n = \frac{1}{2} \cdot \left(\frac{1}{2}\right)^{(n-1)}]

Simplifying this expression gives us the nth term of the sequence:

[a_n = \frac{1}{2^n}]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.