Identify The First 4 Terms In The Geometric Sequence Given By The Explicit Formula F ( N ) = 4 × 2 ( N − 1 ) F(n) = 4 \times 2^{(n-1)} F ( N ) = 4 × 2 ( N − 1 ) .A) 2 , 8 , 32 , 128 2, 8, 32, 128 2 , 8 , 32 , 128 B) 2 , 6 , 10 , 14 2, 6, 10, 14 2 , 6 , 10 , 14 C) 4 , 8 , 16 , 32 4, 8, 16, 32 4 , 8 , 16 , 32 D) 4 , 6 , 8 , 10 4, 6, 8, 10 4 , 6 , 8 , 10

Understanding Geometric Sequences

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The explicit formula for a geometric sequence is given by $f(n) = a \times r^{(n-1)}$, where $a$ is the first term and $r$ is the common ratio.

Given Explicit Formula

The explicit formula given in the problem is $f(n) = 4 \times 2^{(n-1)}$. To identify the first 4 terms in this geometric sequence, we need to substitute the values of $n$ from 1 to 4 into the formula.

Calculating the First 4 Terms

Let's calculate the first 4 terms of the geometric sequence using the given explicit formula.

First Term (n = 1)

$f(1) = 4 \times 2^{(1-1)}$ $f(1) = 4 \times 2^0$ $f(1) = 4 \times 1$ $f(1) = 4$

Second Term (n = 2)

$f(2) = 4 \times 2^{(2-1)}$ $f(2) = 4 \times 2^1$ $f(2) = 4 \times 2$ $f(2) = 8$

Third Term (n = 3)

$f(3) = 4 \times 2^{(3-1)}$ $f(3) = 4 \times 2^2$ $f(3) = 4 \times 4$ $f(3) = 16$

Fourth Term (n = 4)

$f(4) = 4 \times 2^{(4-1)}$ $f(4) = 4 \times 2^3$ $f(4) = 4 \times 8$ $f(4) = 32$

Identifying the Correct Answer

Based on the calculations above, the first 4 terms in the geometric sequence are $4, 8, 16, 32$. Therefore, the correct answer is:

C) $4, 8, 16, 32$

Conclusion

Understanding Geometric Sequences

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The explicit formula for a geometric sequence is given by $f(n) = a \times r^{(n-1)}$, where $a$ is the first term and $r$ is the common ratio.

Frequently Asked Questions

Q1: What is a geometric sequence?

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q2: What is the explicit formula for a geometric sequence?

The explicit formula for a geometric sequence is given by $f(n) = a \times r^{(n-1)}$, where $a$ is the first term and $r$ is the common ratio.

Q3: How do I identify the first term in a geometric sequence?

To identify the first term in a geometric sequence, you need to substitute the value of $n$ into the explicit formula. For example, if the explicit formula is $f(n) = 4 \times 2^{(n-1)}$, then the first term is $f(1) = 4 \times 2^0 = 4$.

Q4: How do I identify the common ratio in a geometric sequence?

To identify the common ratio in a geometric sequence, you need to divide any term by its previous term. For example, if the first term is 4 and the second term is 8, then the common ratio is $8/4 = 2$.

Q5: How do I calculate the nth term in a geometric sequence?

To calculate the nth term in a geometric sequence, you need to substitute the value of $n$ into the explicit formula. For example, if the explicit formula is $f(n) = 4 \times 2^{(n-1)}$, then the nth term is $f(n) = 4 \times 2^{(n-1)}$.

Q6: What is the formula for the sum of the first n terms of a geometric sequence?

The formula for the sum of the first n terms of a geometric sequence is given by $S_n = a \times \frac{1 - r^n}{1 - r}$, where $a$ is the first term and $r$ is the common ratio.

Q7: How do I determine if a sequence is geometric?

To determine if a sequence is geometric, you need to check if each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q8: What are some real-world applications of geometric sequences?

Geometric sequences have many real-world applications, including population growth, compound interest, and music. For example, the population of a city may grow at a rate of 20% per year, which is a geometric sequence.

Conclusion

In this article, we answered some frequently asked questions about geometric sequences. We covered topics such as the explicit formula, identifying the first term, common ratio, and nth term, as well as the formula for the sum of the first n terms and real-world applications. We hope this article has been helpful in understanding geometric sequences.

Additional Resources

• Geometric Sequences Wikipedia Article
• Geometric Sequences Khan Academy Video
• Geometric Sequences Math Is Fun Article