This page focuses on applying the formulas and concepts of geometric sequences to solve various problems. It demonstrates how to find different elements of a sequence given partial information.

One of the key problems presented is:

Example: Calculate q₂, q₃, q₄, q₅ for the sequence: 3/5, 0, 0, 0, 0, 135

The solution process involves recognizing that the zeros in the sequence are placeholders, and the actual sequence is 3/5, 27/5, 81/5, 243/5, 729/5, 135.

Highlight: When solving problems with geometric sequences, it's crucial to identify the actual terms of the sequence, even when some are not explicitly given.

The page also introduces the concept of the middle term in a geometric sequence:

Definition: For a geometric sequence with three terms $an-1, an, an+1$, the middle term $an$ is the geometric mean of the other two terms. This relationship is expressed as an² = an-1 · an+1.

This concept is illustrated with the example: 3, x, 12, where x is the middle term to be found.

The page concludes with more complex problems that require students to apply multiple concepts and formulas to find solutions, reinforcing the practical application of geometric sequence principles.

This final page delves into more complex problems involving geometric sequences, challenging students to apply their knowledge in diverse scenarios.

One notable problem presented is:

Example: Calculate a₁ and q of a geometric sequence where a₆ = 32 and q₁₀ = 2.

This problem requires working backwards from given information about later terms in the sequence to determine the initial term and common ratio.

The page also covers the wzór na sumę nieskończonego ciągu geometrycznego $formula for the sum of an infinite geometric series$:

S∞ = a₁ / $1 - q$ for |q| < 1

Highlight: The sum of an infinite geometric series only converges when the absolute value of the common ratio is less than 1. This is a crucial concept in advanced mathematics and has applications in various fields.

Another significant problem tackled on this page involves finding the nth term of a geometric sequence given its sum and common ratio:

Example: Calculate an of a geometric sequence where Sn = 728 and q = 3.

This problem demonstrates how to use the sum formula in reverse, solving for the nth term when the sum is known.

The page concludes with a comprehensive problem that ties together multiple concepts:

Example: For the sequence an = 7 · 3^$n-1$, calculate q and the first three terms.

This final example reinforces the relationship between the general term formula and the actual terms of the sequence, providing a fitting conclusion to the study of geometric sequences.

This page introduces the concept of geometric sequences and provides essential formulas for working with them. It covers the general form of a geometric sequence and presents key equations for finding specific terms and sums.

The page begins by showing an example of a geometric sequence: 3, 6, 12, 24, 48, 96... This sequence illustrates the defining characteristic of geometric progressions, where each term is a constant multiple of the previous term.

Definition: A geometric sequence is a 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 wzór na n-ty wyraz ciągu geometrycznego $formula for the nth term of a geometric sequence$ is presented:

an = a₁ · q^$n-1$

Where:

• an is the nth term
• a₁ is the first term
• q is the common ratio
• n is the position of the term

Example: For the sequence 3, 6, 12, 24..., we can see that a₁ = 3 and q = 2. Using the formula, we can find a₄ = 3 · 2^$4-1$ = 3 · 8 = 24.

The page also introduces the wzór na sumę ciągu geometrycznego $formula for the sum of a geometric sequence$:

Sn = a₁ · $1 - q^n$ / $1 - q$ for q ≠ 1 Sn = n · a₁ for q = 1

Highlight: The sum formula has two cases depending on whether the common ratio is equal to 1 or not. This is crucial for correctly applying the formula in problem-solving.

The page concludes with several practice problems that apply these formulas in various scenarios, helping students solidify their understanding of geometric sequences.