# If the third term is 3 of a geometric is 36 and the sixth term is 9/2, what is the explicit formula for the sequence ?

The formula will be

Thus:

Hopefully this helps!

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In this example we are given two terms, as follows:

Applying the formula for the general term above:

Square both sides:

Thus our general term for the sequence is:

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To find the explicit formula for the geometric sequence, we can use the formula:

[ a_n = a_1 \times 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 of the sequence

Given that the third term is 36 and the sixth term is ( \frac{9}{2} ), we can form two equations using the formula:

For the third term: [ a_3 = a_1 \times r^{(3-1)} = 36 ]

For the sixth term: [ a_6 = a_1 \times r^{(6-1)} = \frac{9}{2} ]

We can then solve these equations simultaneously to find the values of ( a_1 ) and ( r ). Once we find these values, we can write the explicit formula for the sequence.

Let's solve the equations:

From the third term equation: [ a_1 \times r^2 = 36 ]

From the sixth term equation: [ a_1 \times r^5 = \frac{9}{2} ]

Divide the equation for the sixth term by the equation for the third term to eliminate ( a_1 ): [ \frac{r^5}{r^2} = \frac{\frac{9}{2}}{36} ]

[ r^3 = \frac{1}{8} ]

[ r = \sqrt[3]{\frac{1}{8}} ]

[ r = \frac{1}{2} ]

Substitute ( r = \frac{1}{2} ) into the third term equation to find ( a_1 ): [ a_1 \times (\frac{1}{2})^2 = 36 ]

[ a_1 \times \frac{1}{4} = 36 ]

[ a_1 = 36 \times 4 ]

[ a_1 = 144 ]

Therefore, the explicit formula for the sequence is:

[ a_n = 144 \times (\frac{1}{2})^{(n-1)} ]

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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.

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.

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.

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