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What is the value of the fourth term in a geometric sequence for which [tex]$a_1=30$[/tex] and [tex]$r=\frac{1}{2}$[/tex]?

Sagot :

GinnyAnsweridn
To find the value of the fourth term in a geometric sequence, we can use the formula for the [tex]\(n\)[/tex]th term of a geometric sequence, which is given by:

[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]

Here, [tex]\(a_1\)[/tex] is the first term of the sequence, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number we are looking for.

Given:
- [tex]\(a_1 = 30\)[/tex]
- [tex]\(r = \frac{1}{2}\)[/tex]
- [tex]\(n = 4\)[/tex]

We substitute these values into the formula:

[tex]\[ a_4 = 30 \cdot \left(\frac{1}{2}\right)^{4-1} \][/tex]

Simplify the exponent:

[tex]\[ a_4 = 30 \cdot \left(\frac{1}{2}\right)^3 \][/tex]

Calculate [tex]\(\left(\frac{1}{2}\right)^3\)[/tex]:

[tex]\[ \left(\frac{1}{2}\right)^3 = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8} \][/tex]

So,

[tex]\[ a_4 = 30 \cdot \frac{1}{8} \][/tex]

Multiply the terms:

[tex]\[ a_4 = \frac{30}{8} = 3.75 \][/tex]

Thus, the value of the fourth term in the geometric sequence is:

[tex]\[ \boxed{3.75} \][/tex]
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