Understanding Arithmetic and Geometric Sequences with Summation Examples

Arithmetic Sequence Analysis

• Given the linear function for the arithmetic sequence: u_n = 7n - 2, where n is a natural number.
• The common difference (d) of an arithmetic sequence equals the slope of the linear function.
• Here, the slope is 7, so the common difference d = 7.

Sigma Notation for Arithmetic Sequence

• The summation expression for terms u_1 to u_20 is: [ \sum_{n=1}^{20} (7n - 2) ]
• This represents the sum of the first 20 terms of the arithmetic sequence.

Geometric Sequence Analysis

• The geometric sequence is defined as v_n = 8 * 0.2^{n-1}, with n as a natural number.
• The common ratio (r) is the base of the exponent, which is 0.2.

Summation of Geometric Series

• To find the sum of the first 7 terms, write the summation as: [ \sum_{k=1}^{7} 8 \times 0.2^{k-1} ]
• Substituting each term into the summation formula allows calculation of the series sum.
• Using a math tool or formula, the sum evaluates to 10.0.

Key Takeaways

• The common difference in an arithmetic sequence corresponds to the slope of its linear formula.
• The common ratio in a geometric sequence is the base of the exponential term.
• Sigma notation efficiently expresses sums of sequences.
• Calculating geometric series sums involves substituting terms into the summation and applying the geometric series formula or computational tools.

For a deeper understanding of arithmetic sequences, check out Mastering Sequence and Series: A Comprehensive Guide. If you're interested in practical applications, see Calculating Profits Using Arithmetic and Geometric Sequences. To grasp the basics of arithmetic, refer to Understanding Addition and Subtraction: Basics of Arithmetic. For translating expressions, visit Translating Verbal Expressions into Mathematical Expressions. Lastly, for insights on averages and ratios, explore Understanding Averages, Ratios, and Proportions in Mathematics.