Question

Please help with algebra 2 question! arithmetic sequence

erica is playing a card game with her friend marquis. her points are shown in the following table:

round 1: 30 pts
round 2: 42 pts
round 3: 58.8 pts

marquis’s points in each round are represented by the following sequence:
50, 75, 112.5,…

in the table below, determine whether each row applies to eeica’s points or marquis’s points. select erica, marquis, or neither option.

(answers attached in pic) pls help!

Answer 1

1. neither
2. Marquis
3. Erica
4. Marquis

Explanation:

The matching of exponential functions to table values involves matching both initial values and rate of growth. The value of a geometric series can be calculated from its formula, or can be shown by a calculator or spreadsheet.

Function determination

The starting values of the two sequences are different, as are all of the starting values of the answer choices. Matching starting values is sufficient to identify which sum goes with which player.

The series ...

`\displaystyle S=\sum_(k=1)^n{AB^(k-1)}`

gives the sum of a geometric sequence with first term A and common ratio B.

The first term of Erica's series is A=30. The ratio is B=42/30 = 1.4.

The first term of Marquis's series is 50. The ratio is B=75/50 = 1.5.

Using these values in the series formula, the total score after n rounds will be ...

`\displaystyle \text{Erica:}\quad \sum_(k=1)^n{30(1.4)^(k-1)}\qquad\text{matches Row 3}\\\\\text{Marquis:}\quad \sum_(k=1)^n{50(1.5)^(k-1)}\qquad\text{matches Row 2}`

Total score

The explicit expression for these sums is ...

`S=A(B^n-1)/(B-1)`

Then the total scores after 6 rounds are ...

Erica: 30(1.4^6 -1)/0.4 = 489.7152

Marquis: 50(1.5^6 -1)/0.5 = 1039.0625 . . . . . matches row 4

Table

Then the required table gets filled in ...

1. neither
2. Marquis
3. Erica
4. Marquis

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Additional comment

The values in the attached table are total scores after each round. One must subtract the previous round's total to find the points scored in a given round.

The purpose of showing totals is to permit matching the value seen in Row 4 of the table.