Final answer:
The value of the fourth term in the given geometric sequence, with a first term of 30 and a common ratio of 1/2, is calculated using the formula for the nth term. By plugging the values into the formula and simplifying, the fourth term is found to be 3.75.
Step-by-step explanation:
To find the value of the fourth term in a geometric sequence, we can use the formula for the nth term of a geometric sequence, which is an = a1 × r(n-1). Given that the first term (a1) is 30 and the common ratio (r) is 1/2, we can calculate the fourth term (a4) as follows:
- Identify the first term (a1) and the common ratio (r): a1 = 30, r = 1/2.
- Plug the values into the formula for the nth term: a4 = 30 × (1/2)(4-1).
- Calculate the exponent: (1/2)3 = 1/8.
- Multiply the first term by the result: a4 = 30 × 1/8 = 3.75.
A geometric sequence is a sequence of numbers in which each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).
Given that a₁ = 30 and r = 1/2, we can find the fourth term (a₄) using the formula: a₄ = a₁ * r³, where r³ is the cube of the common ratio.
Substituting the given values, we have: a₄ = 30 * (1/2)³ = 30 * (1/8) = 30/8 = 3.75.
Therefore, the value of the fourth term in the geometric sequence is 3.75.