Understanding Sequences in Profit Calculations
When dealing with profit changes over time, two types of sequences commonly appear:
Arithmetic Sequence
- Profits increase or decrease by the same fixed amount each period.
- The common difference (D) is positive for increases and negative for decreases.
- Example phrases: "increased by $25", "decreased by $25".
Geometric Sequence
- Profits change by a fixed percentage of the previous period.
- The common ratio (r) is calculated as 1 plus or minus the percentage change.
- Example phrases: "increased by 10%", "decreased by 5%", "95% of the previous week".
Example Problems
1. Arithmetic Sequence: Profit Increasing by $25 Weekly
- Initial profit (U1): $400
- Common difference (D): +$25
- Find profit in the 5th week (U5):
- Formula: U5 = U1 + (5 - 1) × D
- Calculation: 400 + 4 × 25 = $500
2. Arithmetic Sequence: Profit Decreasing by $25 Weekly
- Initial profit (U1): $400
- Common difference (D): -$25
- Find profit in the 5th week (U5):
- Formula: U5 = U1 + (5 - 1) × D
- Calculation: 400 + 4 × (-25) = $300
3. Geometric Sequence: Profit Increasing by 10% Weekly
- Initial profit (U1): $400
- Common ratio (r): 1 + 10% = 1.1
- Find profit in the 5th week (U5):
- Formula: U5 = U1 × r^(5 - 1)
- Calculation: 400 × 1.1^4 ≈ $500.86
- Rounded to nearest cent: $500.86
4. Geometric Sequence: Profit Decreasing by 5% Weekly
- Initial profit (U1): $400
- Common ratio (r): 1 - 5% = 0.95
- Find profit in the 5th week (U5):
- Formula: U5 = U1 × r^(5 - 1)
- Calculation: 400 × 0.95^4 ≈ $325.80
5. Geometric Sequence: Profit is 95% of Previous Week
- This is equivalent to a 5% decrease.
- Same calculation as above with r = 0.95.
Key Takeaways
- Identify whether the problem describes an arithmetic or geometric sequence based on wording.
- Use the appropriate formula:
- Arithmetic: Un = U1 + (n - 1) × D
- Geometric: Un = U1 × r^(n - 1)
- Convert percentage increases/decreases to decimal form for calculations.
- Round financial answers to the nearest cent for accuracy.
By applying these principles, you can accurately calculate profits over time in various business scenarios. For a deeper understanding of sequences, consider reading Mastering Sequence and Series: A Comprehensive Guide. If you're interested in the foundational concepts of arithmetic, check out Understanding Addition and Subtraction: Basics of Arithmetic. Additionally, for a broader perspective on ratios and proportions, see Understanding Averages, Ratios, and Proportions in Mathematics.
you word problems increased by the same amount say
means arithmetic sequence common difference D equals this C when you see the word decreased or reduced by the
same amount C the common difference is negative C for geometric
sequence you could see the following words % of the previous one then common ratio is this
a% say times the previous one common ratio is say increased by 8% common ratio is 1 plus
8% decreased or reduced by 8% common ratio is y minus 8% a new t- shop opened last year during
the first week its profit was 400 $ the t- Shop's profit increased by $25 every week what's the t- Shop's
profit in his fifth week increased by $25 this is a arithmetic sequence and a common difference is a
25 we are given usable one = 400 D =
25 us 5 equal usable 1 + 5 - 1 * D
substitution 400 + 4 * 25 equals 500
$ let's go to Second situation the t- Shop's profit decreased by $25 every week was the t- Shop's profit
ye it's fifth week this time decrease byy means aramatic sequence but the common
difference is negative 25 usable 5 = usable 1 + 5 - 1 *
D substitution 400 + 4
* 25 = 300 a new Cafe open at the same time
during the first week his profit was also $400 the Cafe's profit increased by
10% every week was the Cafe's profit in his fifth week increased by
10% we know this is a geometric sequence common ratio r equal 1 + 10% equal
1.1 usable 1 equal 400 so UB 5 = US 1 * R to the 5 - 1's power equals 400 * 1.1 to the 4th power
equals equals 500 8564 for financial problem normally the answer rounded to nearest scent in
second situation the Cafe's profit decreased by 5% every week was the Cafe's profit in
his fifth week decreased by 5% every week means this is a geometric sequence r = 1 -
5% so usable 1 = 400 r equal
95% equals 0.95 so us five
equals 400 * 0.95 to the fource power equals $ 32580 question three the Cafe's profit
was 95% of the previous week was the Cafe's profit in his fifth week was the
95% the previous week means this is a geometric sequence and r equal
95% so U sub five equals 400 time
0.95 to the 4th power equals $325 80 for
Arithmetic sequences involve profits changing by a fixed amount each period, while geometric sequences involve profits changing by a fixed percentage of the previous period. For example, an arithmetic sequence might show profits increasing by $25 each week, whereas a geometric sequence could show profits increasing by 10% each week.
To calculate profits using an arithmetic sequence, use the formula Un = U1 + (n - 1) × D, where U1 is the initial profit, n is the week number, and D is the common difference. For instance, if your initial profit is $400 and it increases by $25 weekly, the profit in the 5th week would be calculated as 400 + (5 - 1) × 25 = $500.
For geometric sequences, use the formula Un = U1 × r^(n - 1), where U1 is the initial profit, r is the common ratio, and n is the week number. For example, if your initial profit is $400 and it increases by 10% weekly, the common ratio would be 1.1, and the profit in the 5th week would be 400 × 1.1^4, which is approximately $500.86.
To determine if a profit change is arithmetic or geometric, look for keywords in the problem. If the profit changes by a fixed amount (e.g., "increased by $25"), it is arithmetic. If it changes by a percentage (e.g., "increased by 10%"), it is geometric.
Rounding financial answers to the nearest cent is crucial for accuracy in financial reporting and transactions. It ensures that calculations reflect real-world currency values, preventing discrepancies in accounting and financial analysis.
Certainly! If your initial profit is $400 and it decreases by 5% each week, the common ratio would be 0.95. To find the profit in the 5th week, you would calculate it as 400 × 0.95^4, which results in approximately $325.80.
Understanding sequences helps businesses forecast profits, analyze trends, and make informed financial decisions. By applying arithmetic and geometric sequences, businesses can predict future earnings based on past performance, which is essential for budgeting and strategic planning.
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