Question

Find an explicit formula for the geometric sequence

120,60,30,15
Note: the first term should be a(1)

Answer 1

Explanation:

The given geometric sequence is: 120, 60, 30, 15.

To find the explicit formula for this sequence, we need to determine the common ratio (r) first. The common ratio is the ratio of any term to its preceding term. Thus,

r = 60/120 = 30/60 = 15/30 = 0.5

Now, we can use the formula for the nth term of a geometric sequence:

a(n) = a(1) * r^(n-1)

where a(1) is the first term of the sequence, r is the common ratio, and n is the index of the term we want to find.

Using this formula, we can find the explicit formula for the given sequence:

a(n) = 120 * 0.5^(n-1)

Therefore, the explicit formula for the given geometric sequence is:

a(n) = 120 * 0.5^(n-1), where n >= 1.

Answer 2

`a_n=120\left((1)/(2)\right)^(n-1)`

Explanation:

An explicit formula is a mathematical expression that directly calculates the value of a specific term in a sequence or series without the need to reference previous terms. It provides a direct relationship between the position of a term in the sequence and its corresponding value.

The explicit formula for a geometric sequence is:

`\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=a_1r^(n-1)$\\\\where:\\\phantom{ww}$\bullet$ $a_1$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}`

Given geometric sequence:

• 120, 60, 30, 15, ...

To find the explicit formula for the given geometric sequence, we first need to calculate the common ratio (r) by dividing a term by its preceding term.

`r=(a_2)/(a_1)=(60)/(120)=(1)/(2)`

Substitute the found common ratio, r, and the given first term, a₁ = 120, into the formula:

`a_n=120\left((1)/(2)\right)^(n-1)`

Therefore, the explicit formula for the given geometric sequence is:

`\boxed{a_n=120\left((1)/(2)\right)^(n-1)}`