Applications of Sequence and Series in Daily Life
These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 12 Mathematics Textbook, Unit 1, Chapter 1.5.
~6 min read- You use an arithmetic model when a quantity changes by a fixed amount each step, like a salary that rises by the same number of birr.
- You use a geometric model when a quantity changes by a fixed percent or ratio each step, like money that grows by interest or a value that loses a percent each year.
- You pick the nth-term formula for a single value, a partial-sum formula for a total, and the sum to infinity for an endless shrinking total.
Key Words
- Arithmetic model: an arithmetic model fits a situation where the same fixed amount is added each step.
- Geometric model: a geometric model fits a situation where the quantity is multiplied by the same ratio each step.
- Growth: growth happens when a quantity increases each period, such as a population that rises by a percent each year.
- Depreciation: depreciation happens when a quantity loses a fixed percent of its value each period.
- Sum to infinity: the sum to infinity is the finite total of an endless geometric process whose steps keep shrinking.
What This Chapter Is About
- This chapter shows how sequences and series describe real situations in money, work, and nature.
- You learn to read a word problem, decide whether it is arithmetic or geometric, and pick the right formula.
Choosing Arithmetic or Geometric
- If the problem adds or subtracts the same fixed amount each step, then it is arithmetic.
- If the problem multiplies by the same ratio, or changes by a fixed percent each step, then it is geometric.
- A fixed birr raise each year is arithmetic, but a fixed percent of interest each year is geometric.
Steps to Solve a Word Problem
- First, write out the first few terms so you can see the pattern clearly.
- Next, decide if the change is a fixed amount, which is arithmetic, or a fixed ratio, which is geometric.
- Then find the first term and the common difference d or common ratio r.
- After that, choose the nth-term formula for one value, the sum formula for a total, or the sum to infinity for an endless shrinking total.
- Finally, check that the answer makes sense, and reject any negative or non-whole value of n.
Real-Life Examples
Money that rises by a fixed amount (arithmetic)
- A worker who starts on 5200 birr a month and gains a fixed 320 birr raise each month forms an arithmetic sequence.
- You use the nth-term formula for the salary in any month, and the sum formula for total earnings over a year.
Money and nature that change by a fixed percent (geometric)
- A bank deposit that grows by 6 percent each year multiplies by 1.06 each time, so the balances form a geometric sequence.
- A town whose population rises by 2.5 percent a year, or a machine that loses one tenth of its value each year, is also geometric.
Figure 1.5: a flower grows 10 cm, then 5 cm, then 2.5 cm, with each year’s growth half of the last, so its total height climbs toward a ceiling of 130 cm but never passes it.
- When the steps keep halving, the total growth is a geometric series with r = 1/2, so the sum to infinity gives the final ceiling.
- This shows how an endless process in nature can still have a clear limit that it never goes beyond.
Which Tool to Use
| What the question asks | Tool to use | Example signal |
|---|---|---|
| A single value at one step | nth-term formula | “salary in the 10th year” |
| A total over many steps | Partial-sum formula Sn | “total earned over 10 years” |
| An endless shrinking total | Sum to infinity S∞ | “final height it never passes” |
| Fixed amount each step | Arithmetic model (d) | “raise of 320 birr a month” |
| Fixed percent each step | Geometric model (r) | “interest of 6 percent a year” |
Common Mistakes to Avoid
- Do not choose arithmetic when the change is a percent, because a percent change is geometric.
- Do not use the sum to infinity unless the ratio lies between −1 and 1, since otherwise the total has no limit.
- Do not mix up the nth-term formula with the sum formula, because one gives a single value and the other gives a total.
- Do not accept a negative or fractional answer for n, because the number of steps must be a positive whole number.
Easy Ways to Remember
- Fixed birr means add, so it is arithmetic; fixed percent means times, so it is geometric.
- One step asks for the nth term, while many steps ask for a sum.
- An endless process that keeps shrinking points you to the sum to infinity.
Quiz
Tap an answer to check it.
1. A salary rises by a fixed 300 birr each month. Which model fits this?
A fixed amount is added each step, so the change is arithmetic.
2. A bank balance grows by 6 percent each year. Which model fits this?
A fixed percent multiplies the balance each year, so the change is geometric.
3. A worker starts on 5200 birr a month with a 320 birr raise each month. What is the salary in the 4th month?
A4 = A1 + (4 − 1)d = 5200 + 3(320) = 5200 + 960 = 6160 birr.
4. A question asks for the total earned over the first 10 years. Which tool do you use?
A total over many steps calls for the partial-sum formula Sn.
5. A flower grows 10 cm, then 5 cm, then 2.5 cm, each year half of the last. What is the total growth in the long run?
This is geometric with G1 = 10 and r = 1/2, so S∞ = 10 ÷ (1 − 1/2) = 20 cm.
Remember: Use an arithmetic model for a fixed amount added each step and a geometric model for a fixed percent or ratio. Pick the nth-term formula for one value, the partial-sum formula for a total, and the sum to infinity for an endless shrinking total, then always check that n is a positive whole number.