12.1 Quiz Review

Quiz Items

1. Sequence Terms and Sum

  • Finding the First Three Terms: Identify the arithmetic sequence or geometric sequence provided and write out the first three terms explicitly.

  • Finding the Last Three Terms: Specify the last three terms of the sequence by determining the value of the last term based on the formula of the sequence.

  • Calculating the Sum: Use the appropriate formula to find the sum of the entire series.

    • Arithmetic Series Sum Formula:
      Sn=racn2(a+l)S_n = rac{n}{2} (a + l)
      where ( S_n ) is the sum of the first ( n ) terms, ( a ) is the first term, ( l ) is the last term, and ( n ) is the number of terms.

    • Geometric Series Sum Formula:
      Sn=arac1−rn1−rS_n = a rac{1 - r^n}{1 - r}
      where ( S_n ) is the sum of the first ( n ) terms, ( a ) is the first term, ( r ) is the common ratio, and ( n ) is the number of terms.

2. Sigma Notation and Value

  • Writing the Sum Using Sigma Notation: Express the identified series using sigma notation ( \ Sum = \sum_{i=1}^{n} f(i) \$ where f(i) is the function representing the terms in the series.

  • Finding its Value: Evaluate the definite sum by substituting values into the sigma notation and simplifying.

3. Finding ( n ) for an Arithmetic Series

  • Determine ( n ) in an arithmetic series where necessary parameters such as the first term, common difference, and last term are given.

  • Use the formula:
    n=racl−ad+1n = rac{l - a}{d} + 1
    where ( n ) is the number of terms, ( l ) is the last term, ( a ) is the first term, and ( d ) is the common difference.

4. Sum of the Series

  • Calculate the sum of the series by applying the respective formula based on whether the series is arithmetic or geometric.

5. Sigma Notation for the Sum

  • Expressing the Sum with Sigma Notation: Similar to item 2, write the series in sigma notation and calculate the sum from it.

6. (Unspecified Item)
7. Finding ( n ) for a Geometric Series

  • Derive the value of ( n ) based on the first term, common ratio, and any given parameters of the geometric series.

  • Formula used:
    n=racextlog(Sn)−extlog(a)extlog(r)n = rac{ ext{log}(S_n) - ext{log}(a)}{ ext{log}(r)}
    where ( S_n ) is the sum of the first ( n ) terms, ( a ) is the first term, and ( r ) is the common ratio.

8. Alternating Series

  • Finding the Sum: Given the series: 108−72+48−32+ext…108 - 72 + 48 - 32 + ext{…}

    • Note the common difference or ratio to derive the series type and evaluate its sum.

9. Stadium Seating Problem

  • Rows and Seats: A stadium with 75 rows where 10 seats are in the first row and 15 seats in the second row follows an arithmetic progression.

    • Finding Seats in the Top Row: Determine the number of seats in the top row using the formula for the nth term of an arithmetic sequence:
      an=a+(n−1)da_n = a + (n-1)d
      where ( a ) is the first term, ( d ) is the common difference, and ( n ) is the row number.

  • Total Seats in the Stadium: Sum the total seats using the arithmetic series formula, evaluating from row 1 to row 75.

10. Savings Account Scenario

  • Investment and Interest Calculation: Emily invests $500 monthly at an interest of 0.25%.

    • Total Investment After 3 Years: Calculate the total number of months in 3 years, which equals 36 months.

    • Formula for Future Value of an Annuity:
      FV=Pimesrac(1+r)n−1rFV = P imes rac{(1 + r)^n - 1}{r}
      where ( P ) is the monthly investment, ( r ) is the monthly interest rate, and ( n ) is the total number of investments.

    • Will she have enough?: Compare the calculated future value with the price of the car, $20,000.