Comprehensive Notes on Sigma Notation, Series Expansions, and Financial Applications

EXPANSION AND EVALUATION OF SIGMA NOTATION

Sigma notation is a concise method used to write long sums. The Greek capital letter sigma, $\sum$, indicates a summation, where an index variable takes integer values from a lower limit to an upper limit.

• Evaluation Example 1A:

  • Problem: Expand the sigma notation $\sum_{n=1}^{5} 6n$ and calculate the final sum.   
  • Expansion Step: Substitute values of $n$ from $1$ to $5$ into the expression $6n$:          $6(1) + 6(2) + 6(3) + 6(4) + 6(5)$
  • Term-by-Term Evaluation: Multiply each index value by $6$:          $6 + 12 + 18 + 24 + 30$
  • Final Sum Calculation: Add the calculated terms together:          $6 + 12 + 18 + 24 + 30 = 90$

• Evaluation Example 1B:

  • Problem: Expand the sigma notation $\sum_{n=1}^{4} (2n - 1)$ and calculate the final sum.   
  • Expansion Step: Substitute values of $n$ from $1$ to $4$ into the expression $2n - 1$:          $[2(1) - 1] + [2(2) - 1] + [2(3) - 1] + [2(4) - 1]$
  • Simplification of Individual Terms: Compute the inside of each bracket:          $(2 - 1) + (4 - 1) + (6 - 1) + (8 - 1)$
  • Term-by-Term Evaluation:$1 + 3 + 5 + 7$
  • Final Sum Calculation: Add the calculated odd integer terms together:          $1 + 3 + 5 + 7 = 16$

CONVERTING ARITHMETIC AND GEOMETRIC SERIES INTO SIGMA NOTATION

To write a given numerical series using sigma notation, identify the pattern for the general term $a_n$, determine the starting index limit $n=1$, and find the total number of terms for the upper limit.

• Problem 3: Write the series $5 + 10 + 15 + 20 + 25$ in sigma notation.   

  • Pattern Recognition: Each term is a consecutive positive multiple of $5$:          $5(1) + 5(2) + 5(3) + 5(4) + 5(5)$
  • General Term Formula: $a_n = 5n$
  • Index Limits: Starts at $n = 1$ and ends at $n = 5$
  • Sigma Notation Result:$\sum_{n=1}^{5} 5n$

• Problem 4: Write the series $3 + 6 + 9 + 12$ in sigma notation.   

  • Pattern Recognition: Each term is a consecutive positive multiple of $3$:          $3(1) + 3(2) + 3(3) + 3(4)$
  • General Term Formula: $a_n = 3n$
  • Index Limits: Starts at $n = 1$ and ends at $n = 4$
  • Sigma Notation Result:$\sum_{n=1}^{4} 3n$

• Problem 5: Write the series $1 + 4 + 7 + 10$ in sigma notation.   

  • Pattern Recognition: An arithmetic sequence starting at $1$ with a common difference of $d = 3$:          $[3(1) - 2] + [3(2) - 2] + [3(3) - 2] + [3(4) - 2]$
  • General Term Formula: $a_n = 3n - 2$
  • Index Limits: Starts at $n = 1$ and ends at $n = 4$
  • Sigma Notation Result:$\sum_{n=1}^{4} (3n - 2)$

• Problem 6: Write the series $1 + 3 + 5 + 7 + 9$ in sigma notation.   

  • Pattern Recognition: An arithmetic sequence of consecutive positive odd integers with a common difference of $d = 2$:          $[2(1) - 1] + [2(2) - 1] + [2(3) - 1] + [2(4) - 1] + [2(5) - 1]$
  • General Term Formula: $a_n = 2n - 1$
  • Index Limits: Starts at $n = 1$ and ends at $n = 5$
  • Sigma Notation Result:$\sum_{n=1}^{5} (2n - 1)$

• Problem 7: Write the series $5 + 7 + 9 + 11 + 13$ in sigma notation.   

  • Pattern Recognition: An arithmetic sequence starting at $5$ with a common difference of $d = 2$:          $a_1 = 5$$a_n = a_1 + (n - 1)d = 5 + (n - 1)2 = 2n + 3$
  • Expanded Substitution Check:$[2(1) + 3] + [2(2) + 3] + [2(3) + 3] + [2(4) + 3] + [2(5) + 3]$
  • Index Limits: Starts at $n = 1$ and ends at $n = 5$
  • Sigma Notation Result:$\sum_{n=1}^{5} (2n + 3)$

• Problem 8: Write the series $1 + 3 + 9 + 27 + 81$ in sigma notation.   

  • Pattern Recognition: A geometric sequence with a first term of $a_1 = 1$ and a common ratio of $r = 3$:          $a_n = a_1 \times r^{n-1} = 1 \times 3^{n-1} = 3^{n-1}$
  • Expanded Substitution Check:$3^{1-1} + 3^{2-1} + 3^{3-1} + 3^{4-1} + 3^{5-1} = 3^0 + 3^1 + 3^2 + 3^3 + 3^4$
  • Index Limits: Starts at $n = 1$ and ends at $n = 5$
  • Sigma Notation Result:$\sum_{n=1}^{5} 3^{n-1}$

APPLICATION OF SEQUENCE AND SERIES IN FINANCIAL PROBLEMS

Sequences and series provide the fundamental mathematical foundation for analyzing structured financial problems involving repetitive payments, interest calculations, compound growth, and debt amortization over time.

• Core Financial Concepts Modeled by Sequences:
  • Simple Interest: Formulates as an arithmetic sequence where a constant interest amount is added per period.   
  • Compound Interest: Formulates as a geometric sequence where principal and accumulated interest grow by a multiplicative growth factor $(1 + i)$ per period.   
  • Annuities and Amortization: Modeled via finite geometric series to evaluate accumulated future values or present discounted values of regular periodic cash flows.