Identifying the Sequence Type

Oct 13, 2025 by pinkandtrash.com 34 views

Understanding the Problem

When dealing with sequences, it's often necessary to calculate the sum of a specific number of terms. In this case, we're given the first three terms of a sequence: 6400, 800, and 100. We're asked to find the sum of the first eight terms of this sequence.

Identifying the Sequence Type

The given sequence appears to be a geometric sequence, where each term is obtained by multiplying the previous term by a fixed constant. In this case, the common ratio (r) can be found by dividing the second term by the first term:

r = 800 / 6400 = 1/8

Calculating the Sum of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be calculated using the formula:

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

where:

S_n is the sum of the first n terms
a is the first term
r is the common ratio
n is the number of terms

Applying the Formula

Using the given values, we can plug in the values into the formula:

a = 6400 r = 1/8 n = 8

Substituting these values into the formula, we get:

S_8 = 6400 * (1 - (1/8)^8) / (1 - 1/8)

Evaluating the Expression

To evaluate this expression, we need to follow the order of operations (PEMDAS):

Calculate the exponent: (1/8)^8 = 1/8^8
Calculate the denominator: 1 - 1/8 = 7/8
Calculate the numerator: 1 - (1/8)^8 = 1 - 1/8^8
Multiply the numerator and denominator: 6400 * (1 - 1/8^8) / (7/8)

Simplifying the Expression

To simplify the expression, we can multiply the numerator and denominator by 8 to eliminate the fraction in the denominator:

S_8 = 6400 * 8 * (1 - 1/8^8) / (7 * 8)

Final Calculation

Now we can simplify the expression further:

S_8 = 51200 * (1 - 1/8^8) / 7

Conclusion

The correct formula to calculate the sum of the first eight terms of the sequence is:

S_8 = 51200 * (1 - 1/8^8) / 7

This formula uses the formula for the sum of a geometric sequence, and it takes into account the given values of the first term, common ratio, and number of terms.

Comparison with the Given Options

Let's compare the derived formula with the given options:

A. S_8 = 6400 * (1 - (1/8)^8) / (1 - 1/8)

B. ...

The derived formula matches option A.

Final Answer

The correct answer is option A.

Additional Information

The sequence is a geometric sequence with a common ratio of 1/8. The sum of the first eight terms can be calculated using the formula:

S_8 = 51200 * (1 - 1/8^8) / 7

This formula can be used to calculate the sum of any number of terms in a geometric sequence, as long as the first term, common ratio, and number of terms are known.

Real-World Applications

Geometric sequences have many real-world applications, such as:

Compound interest: The formula for the sum of a geometric sequence can be used to calculate the future value of an investment with compound interest.
Population growth: The formula for the sum of a geometric sequence can be used to model population growth in a population with a constant birth rate.
Music: The formula for the sum of a geometric sequence can be used to calculate the frequency of a musical note.

Conclusion

In conclusion, the sum of the first eight terms of the sequence can be calculated using the formula:

S_8 = 51200 * (1 - 1/8^8) / 7

This formula uses the formula for the sum of a geometric sequence, and it takes into account the given values of the first term, common ratio, and number of terms.

Understanding Geometric Sequences

A geometric sequence is a type of sequence where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. The formula for the sum of a geometric sequence is:

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

where:

S_n is the sum of the first n terms
a is the first term
r is the common ratio
n is the number of terms

Q&A

Q: What is the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence is:

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

Q: What is the common ratio (r) in a geometric sequence?

A: The common ratio (r) is the fixed constant that is multiplied by each term to obtain the next term in the sequence.

Q: How do I calculate the sum of the first n terms of a geometric sequence?

A: To calculate the sum of the first n terms of a geometric sequence, you can use the formula:

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

where:

S_n is the sum of the first n terms
a is the first term
r is the common ratio
n is the number of terms

Q: What is the difference between a geometric sequence and an arithmetic sequence?

A: A geometric sequence is a type of sequence where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. An arithmetic sequence, on the other hand, is a type of sequence where each term is obtained by adding a fixed constant to the previous term.

Q: Can I use the formula for the sum of a geometric sequence to calculate the sum of an arithmetic sequence?

A: No, the formula for the sum of a geometric sequence cannot be used to calculate the sum of an arithmetic sequence. The formula for the sum of an arithmetic sequence is different and is given by:

S_n = n/2 * (a + l)

where:

S_n is the sum of the first n terms
n is the number of terms
a is the first term
l is the last term

Q: How do I determine the common ratio (r) of a geometric sequence?

A: To determine the common ratio (r) of a geometric sequence, you can divide any term by the previous term. For example, if the first two terms of the sequence are 2 and 6, then the common ratio (r) is:

r = 6 / 2 = 3

Q: Can I use the formula for the sum of a geometric sequence to calculate the sum of a finite sequence?

A: Yes, the formula for the sum of a geometric sequence can be used to calculate the sum of a finite sequence. Simply plug in the values of the first term, common ratio, and number of terms into the formula.

Q: What is the significance of the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence has many real-world applications, such as calculating the future value of an investment with compound interest, modeling population growth, and calculating the frequency of a musical note.

Q: Can I use the formula for the sum of a geometric sequence to calculate the sum of an infinite sequence?

A: No, the formula for the sum of a geometric sequence cannot be used to calculate the sum of an infinite sequence. The sum of an infinite geometric sequence is given by:

S = a / (1 - r)

where:

S is the sum of the infinite sequence
a is the first term
r is the common ratio

Q: How do I determine the sum of an infinite geometric sequence?

A: To determine the sum of an infinite geometric sequence, you can use the formula:

S = a / (1 - r)

where:

S is the sum of the infinite sequence
a is the first term
r is the common ratio

Note that the sum of an infinite geometric sequence is only defined if the absolute value of the common ratio (r) is less than 1.