Comprehensive Guide to Geometric Sequence Problems and Solutions

Administrative Guidelines and Submission Requirements

• All mathematical exercises must be completed on a whole sheet of paper.

• The document must include the current date written at the top.

• A signature from a parent or guardian is required on the paper to validate the work.

• The final output should be presented in a table format containing the sequence, the solution process, the first term ($a_1$), the common ratio ($r$), and the final calculated answer.

Core Mathematical Definition: Geometric Sequence Formula

• A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$).

• The formula to find the $n^{th}$ term ($a_n$) of a geometric sequence is defined as:     $a_n = a_1 \times r^{n-1}$

• Where:

  • $a_n$ represents the term at position $n$.

  • $a_1$ represents the first term in the sequence.

  • $r$ represents the common ratio between consecutive terms.

  • $n$ represents the position of the term in the sequence.

Analysis of Sequence 1: Calculating the 7th Term

• Given Sequence Details:

  • Sequence: $4, 12, 36, 108, \dots$

  • Target: $7^{th}$ term

• Identification of Variables:

  • $a_1 = 4$

  • $r = \frac{12}{4} = 3$

  • $n = 7$

• Step-by-Step Solution:

  • Substitute values into the geometric sequence formula: $a_7 = 4 \times 3^{7-1}$

  • Simplify the exponent: $a_7 = 4 \times 3^6$

  • Calculate the power of the ratio: $3^6 = 729$

  • Multiply by the first term: $4 \times 729 = 2916$

• Final Answer:

  • $a_7 = 2916$

Analysis of Sequence 2: Calculating the 10th Term

• Given Sequence Details:

  • Sequence: $-3, 6, -12, 24, \dots$

  • Target: $10^{th}$ term

• Identification of Variables:

  • $a_1 = -3$

  • $r = \frac{6}{-3} = -2$

  • $n = 10$

• Step-by-Step Solution:

  • Substitute values into the formula: $a_{10} = -3 \times (-2)^{10-1}$

  • Simplify the exponent: $a_{10} = -3 \times (-2)^9$

  • Calculate the power of the ratio: $(-2)^9 = -512$ (Note: an odd exponent preserves the negative sign)

  • Multiply by the first term: $-3 \times (-512) = 1536$

• Final Answer:

  • $a_{10} = 1536$

Analysis of Sequence 3: Determining the 11th Term

• Given Sequence Details:

  • Sequence: $2187, 729, 243, 81, \dots$

  • Target: $11^{th}$ term

• Identification of Variables:

  • $a_1 = 2187$

  • $r = \frac{729}{2187} = \frac{1}{3}$

  • $n = 11$

• Step-by-Step Solution:

  • Substitute values into the formula: $a_{11} = 2187 \times (\frac{1}{3})^{11-1}$

  • Simplify the exponent: $a_{11} = 2187 \times (\frac{1}{3})^{10}$

  • Express $2187$ as a power of $3$: $2187 = 3^7$

  • Calculate the power of the ratio: $(\frac{1}{3})^{10} = \frac{1}{59049}$

  • Multiply: $a_{11} = 3^7 \times \frac{1}{3^{10}} = \frac{1}{3^3}$

  • Solve the fraction: $a_{11} = \frac{1}{27}$

• Final Answer:

  • $a_{11} = \frac{1}{27}$

Analysis of Sequence 4: Determining the 100th Term

• Given Sequence Details:

  • Sequence: $-5, 5, -5, 5, \dots$

  • Target: $100^{th}$ term

• Identification of Variables:

  • $a_1 = -5$

  • $r = \frac{5}{-5} = -1$

  • $n = 100$

• Step-by-Step Solution:

  • Substitute values into the formula: $a_{100} = -5 \times (-1)^{100-1}$

  • Simplify the exponent: $a_{100} = -5 \times (-1)^{99}$

  • Evaluate the power: Because $99$ is an odd integer, $(-1)^{99} = -1$

  • Multiply by the first term: $-5 \times (-1) = 5$

• Final Answer:

  • $a_{100} = 5$

Analysis of Sequence 5: Determining the 9th Term

• Given Sequence Details:

  • Sequence: $-2, -4, -8, -16, \dots$

  • Target: $9^{th}$ term

• Identification of Variables:

  • $a_1 = -2$

  • $r = \frac{-4}{-2} = 2$

  • $n = 9$

• Step-by-Step Solution:

  • Substitute values into the formula: $a_9 = -2 \times 2^{9-1}$

  • Simplify the exponent: $a_9 = -2 \times 2^8$

  • Calculate the power of the ratio: $2^8 = 256$

  • Multiply by the first term: $-2 \times 256 = -512$

• Final Answer:

  • $a_9 = -512$