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
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