Question

ence below, find the common ratio, zeroth term
t(n) common ratio=
10
20 zeroth term=
40 equation=
-128
tion to find the nth term.
term.
(2)^n

Explanation:

Step1: Find the common ratio

The common ratio $r$ of a geometric - sequence is found by dividing a term by its previous term. Given the sequence $t(n)$ with terms $10,20,40,\cdots$, we use the formula $r=\frac{t(n + 1)}{t(n)}$. Taking $t(1)=10$ and $t(2)=20$, then $r=\frac{20}{10}=2$.

Step2: Find the zeroth - term

The general formula for a geometric sequence is $t(n)=t(0)\times r^{n}$, where $t(n)$ is the $n$th term, $t(0)$ is the zeroth - term, and $r$ is the common ratio. We know $t(1) = 10$ and $r = 2$. Substituting into the formula $t(1)=t(0)\times r^{1}$, we get $10=t(0)\times2$. Solving for $t(0)$, we have $t(0)=\frac{10}{2}=5$.

Step3: Find the equation for the $n$th term

The general formula for a geometric sequence is $t(n)=t(0)\times r^{n}$. Since $t(0) = 5$ and $r = 2$, the equation for the $n$th term is $t(n)=5\times2^{n}$.

Answer:

Common Ratio = 2
Zeroth Term = 5
Equation = $t(n)=5\times2^{n}$