f the first 4 terms of a geometric sequence are $$ \{7,21,63,189\} $$, then ormula for the $$ n^{ ext {th }} $$ term in the sequence is

Question

f the first 4 terms of a geometric sequence are $$ \{7,21,63,189\} $$, then ormula for the $$ n^{	ext {th }} $$ term in the sequence is

UpStudy AI · Accepted Answer

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio $$ r $$. The general formula for the $$ n^{\text{th}} $$ term of a geometric sequence is:

$$
a_n = a_1 \times r^{n-1}
$$

where:
- $$ a_1 $$ is the first term.
- $$ r $$ is the common ratio.
- $$ n $$ is the term number.

Given the first four terms of the sequence: $$ \{7, 21, 63, 189\} $$, let's determine the common ratio $$ r $$:

$$
r = \frac{21}{7} = 3
$$

Now, using the formula for the $$ n^{\text{th}} $$ term:

$$
a_n = 7 \times 3^{n-1}
$$

**Verification:**
- For $$ n = 1 $$: $$ a_1 = 7 \times 3^{0} = 7 $$
- For $$ n = 2 $$: $$ a_2 = 7 \times 3^{1} = 21 $$
- For $$ n = 3 $$: $$ a_3 = 7 \times 3^{2} = 63 $$
- For $$ n = 4 $$: $$ a_4 = 7 \times 3^{3} = 189 $$

All terms match the given sequence.

**Final Formula:**

$$
a_n = 7 \times 3^{n-1}
$$

**Answer:**  
After simplifying, the nth term is 7 multiplied by threeⁿ⁻¹. Thus,

aₙ = 7 · 3^{ n−1 }
The formula for the $$ n^{\text{th}} $$ term in the sequence is $$ a_n = 7 \times 3^{n-1} $$.

f the first 4 terms of a geometric sequence are \( \{7,21,63,189\} \), then ormula for the \( n^{\text {th }} \) term in the sequence is

Solución de tutoría real

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Algebra Questions