https://file.aiquickdraw.com/mathbot/file/1726260213560ec3gl5ic.jpg

The problem involves a geometric sequence with the first term \(a\) and common ratio \(r\). We have the following conditions:

1. The third term is 20.
2. The fifth term is 12.8.
3. \(r = 0.8\) and \(a = 31.25\).

We need to find the smallest \(n\) such that the sum of the first \(n\) terms is greater than 156.

**Solution:**

Given that \(a = 31.25\) and \(r = 0.8\), the \(n\)-th term of the geometric sequence can be expressed as:
\[ a_n = ar^{n-1} = 31.25 \times 0.8^{n-1} \]

The sum \(S_n\) of the first \(n\) terms of a geometric sequence is given by:
\[ S_n = a \frac{1-r^n}{1-r} \]

Plugging in the values:
\[ S_n = 31.25 \frac{1-0.8^n}{1-0.8} = 31.25 \frac{1-0.8^n}{0.2} = 156.25 (1-0.8^n) \]

We need to find the smallest \(n\) such that:
\[ 156.25 (1-0.8^n) > 156 \]

Let's solve for \(n\).

\[ 1 - 0.8^n > \frac{156}{156.25} = 0.9984 \]

\[ 0.8^n < 1 - 0.9984 = 0.0016 \]

Now, we'll calculate \(n\) by taking logarithms:
\[ \log(0.8^n) < \log(0.0016) \]
\[ n \log(0.8) < \log(0.0016) \]
\[ n > \frac{\log(0.0016)}{\log(0.8)} \]

Let's calculate \(n\).The smallest possible value of \(n\) such that the sum of the first \(n\) terms of the geometric sequence is greater than 156 is \(n = 29\).

Would you like more details or have any other questions?

### Further Questions:
1. How would the smallest \(n\) change if the common ratio \(r\) was different?
2. What happens to the sum \(S_n\) as \(n\) approaches infinity for a geometric sequence with \(r < 1\)?
3. How do you find the sum of an infinite geometric series and what conditions must be met for it to converge?
4. Can you calculate the sum of the first 30 terms to confirm that it exceeds 156?
5. What would be the first term if the third term is known and the common ratio changes?

### Tip:
Remember that logarithmic calculations are used to solve inequalities involving exponential terms, especially useful in sequences and series problems.

A geometric sequence has first term a and common ratio r, where r > 0. Given that the 3rd term is 20 and the 5th term is 12.8, it can be shown that r = 0.8 and a = 31.25. Given that the sum of the first n terms of this sequence is greater than 156, find the smallest possible value of n.

This problem involves a geometric sequence with the first term a = 31.25 and common ratio r = 0.8. Learn how to calculate the smallest number of terms needed for the sum to exceed 156. The solution involves key formulas for sequences and logarithmic calculations, suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation