Calculate s12 for the geometric series 4+10+25 – In the realm of mathematics, geometric series captivate with their intriguing patterns and practical applications. This article embarks on an exploration of the geometric series 4 + 10 + 25, delving into the intricacies of calculating its 12th partial sum, S12.

Join us as we unravel the secrets of this captivating sequence.

1. Sum of the First n Terms of a Geometric Series

The sum of the first n terms of a geometric series is given by the formula:

S _n= a(1 – r ⁿ) / (1 – r)

where a is the first term, r is the common ratio, and n is the number of terms.

For example, to calculate the sum of the first 4 terms of the series 4+10+25, we have:

a = 4

r = 10/4 = 2.5

n = 4

So, the sum of the first 4 terms is:

S ₄= 4(1 – 2.5 ⁴) / (1 – 2.5) = 106.8

2. Convergence and Divergence of a Geometric Series

A geometric series is convergent if the absolute value of the common ratio is less than 1, and divergent otherwise.

For the series 4+10+25, the common ratio is 2.5, which is greater than 1. Therefore, the series is divergent.

3. Applications of Geometric Series

Geometric series have numerous applications in various fields, including:

• Finance: calculating compound interest, present value, and future value.
• Science: modeling population growth, radioactive decay, and the cooling of objects.
• Engineering: analyzing electrical circuits and fluid flow.

4. Properties of Geometric Series: Calculate S12 For The Geometric Series 4+10+25

Geometric series have several useful properties, including:

• The sum of a geometric series is equal to the first term divided by 1 minus the common ratio, if the common ratio is less than 1.
• The product of a geometric series is equal to the first term times the common ratio raised to the power of n-1.
• The nth term of a geometric series is equal to the first term times the common ratio raised to the power of n-1.

5. Comparison of Geometric Series

Geometric series are similar to arithmetic series, but they have some key differences:

• Geometric series have a common ratio, while arithmetic series have a common difference.
• Geometric series can be convergent or divergent, while arithmetic series are always convergent.

6. Examples and Exercises

Additional examples of geometric series include:

• 1 + 2 + 4 + 8 + 16
• 100 + 50 + 25 + 12.5 + 6.25

Exercises:

• Calculate the sum of the first 5 terms of the series 3 + 6 + 12 + 24 + 48.
• Determine whether the series 1 + 3 + 9 + 27 + 81 is convergent or divergent.

Popular Questions

What is the formula for calculating the 12th partial sum of a geometric series?

The formula for the nth partial sum of a geometric series with first term a and common ratio r is Sn = a(1 – r^n) / (1 – r), where n is the number of terms.

How do I determine if a geometric series is convergent or divergent?

A geometric series is convergent if |r| < 1 and divergent if |r| >= 1.

What are some real-world applications of geometric series?

Geometric series find applications in finance (compound interest), physics (radioactive decay), and computer science (binary search).