Geometric sequences

• In a geometric sequence where the first term is 2 and the common ratio is 4, what is the 4th term?

  The 4th term is 128, calculated using $a_4 =$$2 \cdot 4^{4-1} =$$2 \cdot 64 =$$128$.
• What are the steps to find the common ratio in a geometric sequence?

  1. Choose any two consecutive terms in the sequence. 2. Divide the later term by the earlier term.
• What is the process to calculate the nth term using the general term formula?


  1. Identify the first term ($a_1$) and the common ratio (r). 2. Plug these values into the formula: $a_n =$$a_1 \cdot r^{n-1}$. 3. Simplify the exponent and calculate the result.
• What is an example of a geometric sequence?

  - The sequence 2, 6, 18, 54, ... is a geometric sequence with a common ratio of 3.
• How does the common ratio affect the terms of a geometric sequence?

  - A larger common ratio results in terms that grow more quickly. - A common ratio less than 1 results in terms that decrease.
• What is the general term formula for a geometric sequence?

  $a_n =$$a_1 \cdot r^{n-1}$
• In a geometric sequence, what do the variables $a_1$, $r$, and $n$ represent?


  $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.
• How do you calculate the 4th term of a geometric sequence with first term 2 and common ratio 4?

  Use the formula $a_4 =$$2 \cdot 4^{4-1} =$$128$.
• What is the 4th term of the geometric sequence where the first term is 2 and the common ratio is 4?

  128
• What are the formulas for the sum of a geometric sequence?

  1. For a finite geometric sequence (n terms): $S_n =$$\frac{a_1(1-r^n)}{1-r}$ (when $r \neq 1$) 2. For an infinite geometric sequence with $|r| < 1$: $S_{\infty} =$$\frac{a_1}{1 - r}$
• How do you find the sum of the first 4 terms of the sequence 2, 6, 18, 54?

  Use the formula $S_4 =$$\frac{2(1-3^4)}{1-3} =$$80$.
• What is the sum of the first 4 terms of the geometric sequence 2, 6, 18, 54?

  80
• What is a geometric sequence?

  A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
• In the sequence 2, 6, 18, 54, what is the common ratio?

  The common ratio is 3, as each term is obtained by multiplying the previous term by 3.
• How do you find the common ratio in a geometric sequence?

  To find the common ratio, divide any term by the previous term.
• What is the common ratio in a geometric sequence?

  The common ratio is the fixed, non-zero number by which each term in a geometric sequence is multiplied to get the next term.