IB Maths AI HLSequences & SeriesPaper 1 & 2interest & growth~7 min read
Applications of Sequences & Series
Worded problems rarely say “arithmetic” or “geometric” — you have to spot which model fits. A fixed amount added each time is arithmetic; a fixed factor multiplied each time is geometric. Once you have chosen, the standard formulae do the rest.
📘 What you need to know
A quantity changing by a fixed amount added or subtracted → use an arithmetic model.
A quantity changing by a fixed factor multiplied each time → use a geometric model.
Simple interest, stacking and pattern problems are arithmetic.
Compound interest, population growth and decay are geometric.
For a percentage change, the ratio is r = 1 + (percent ÷ 100) for growth, 1 − (percent ÷ 100) for decay.
Decide if the question wants a single term (un) or a running total (Sn).
Recognising an arithmetic model
Use an arithmetic model whenever a quantity changes by the same amount added or subtracted each step. The starting value is u1 and the fixed change is the common difference d.
Typical contexts include simple interest, cups stacked in a tapering pile, trees planted in fixed batches, and dots counted in a growing pattern.
The test is the type of change: a constant amount added points to arithmetic; a constant factor multiplied points to geometric.
Arithmetic model — fixed amount addedun = u1 + (n − 1)dSn = n2(2u1 + (n − 1)d)use a term for one value, a sum for a cumulative total
Recognising a geometric model
Use a geometric model whenever a quantity is repeatedly multiplied or divided by the same factor. The starting value is u1 and the fixed factor is the common ratio r.
Typical contexts include compound interest, population and bacterial growth, and radioactive decay.
Geometric model — fixed factor multipliedun = u1rn−1Sn = u1(rn − 1)r − 1for an x% change: r = 1 + x/100 (growth) or 1 − x/100 (decay)
Interest, growth and decay
Simple interest adds the same amount each year — an arithmetic model. Compound interest multiplies by the same factor each year — a geometric model. Growth and decay work the same way: a percentage change per period gives a common ratio.
Watch the percentage: a 6% rise means multiply by 1.06; a 6% fall means multiply by 0.94. Convert the percentage into a ratio before using any formula.
🧠Recipe — tackling an applications problem
Identify the changing quantity and write down its starting value.
Decide the model — a fixed amount added is arithmetic; a fixed factor multiplied is geometric.
State u1 and d or r — for a percentage, turn it into a ratio first.
Choose term or sum — un for a single value, Sn for a cumulative total.
Answer in context — add units, round sensibly and check the result is realistic.
Worked examples
WE 1
Arithmetic: a stacking pattern
Tins are stacked in rows. The bottom row has 28 tins, and each row above has 3 fewer tins. How many tins are in the 8th row?
fixed amount removed each row ⇒ arithmeticu1 = 28, d = −3find the 8th termu8 = 28 + (8 − 1)(−3) = 28 − 217 tinsa decreasing pattern just means a negative common difference.
WE 2
Arithmetic: reaching a total
Maria saves money each week. In week 1 she saves $8, and each week she saves $3 more than the week before. In which week do her total savings first reach at least $200?
fixed amount added ⇒ arithmetic; u1 = 8, d = 3a total ⇒ use Sn; set Sn = 200n2(2(8) + (n − 1)(3)) = 2003n2 + 13n − 400 = 0 ⇒ n ≈ 9.58check whole weeks either sideS9 = 180 (< 200), S10 = 215 (≥ 200)week 10a non-integer n means you test the whole numbers on each side.
WE 3
Simple interest
$2000 is invested at 5% simple interest per year — the interest each year is 5% of the original $2000. (a) Find the interest earned each year. (b) Find the total value of the investment after 6 years.
(a) simple interest: the same amount each yearinterest = 5% of 2000 = 0.05 × 2000 = $100(b) $100 added each year ⇒ arithmeticvalue = 2000 + 6 × 100 = 2000 + 600(a) $100 per year · (b) $2600simple interest adds a constant — the value grows arithmetically.
WE 4
Compound interest
$3000 is invested at 4% compound interest per year. Find the value of the investment after 5 years, to the nearest dollar.
compound interest: multiply by a fixed factor ⇒ geometric4% growth ⇒ r = 1 + 0.04 = 1.04value after 5 years3000 × 1.045 = 3000 × 1.2167= 3649.96…≈ $3650compound interest multiplies each year, so it grows geometrically.
WE 5
Geometric: a spreading total
A post is shared by 5 people on day 1. Each day, the number of people who newly share it is triple the previous day’s. (a) How many people newly share it on day 6? (b) Find the total number of shares over the first 6 days.
tripled each day ⇒ geometric; u1 = 5, r = 3(a) a single day ⇒ use the term formulau6 = 5 × 35 = 5 × 243 = 1215(b) a total ⇒ use the sum formulaS6 = 5(36 − 1)3 − 1 = 5(728)2(a) 1215 · (b) 1820“on day 6” wants a term; “total over 6 days” wants a sum.
WE 6
Full question: radioactive decay
A radioactive sample has a mass of 80 mg and loses 15% of its mass each year. (a) Explain why the masses form a geometric sequence and state the common ratio. (b) Find the mass after 4 years. (c) Find the mass after 10 years, to 1 decimal place.
(a) each year the mass keeps 85% of its valuemultiplied by the same factor ⇒ geometric, r = 0.85(b) mass after n years = 80 × 0.85n80 × 0.854 = 80 × 0.522 = 41.76(c) after 10 years80 × 0.8510 = 15.749…(a) r = 0.85 · (b) 41.8 mg · (c) 15.7 mga 15% loss means keeping 85%, so the decay ratio is 0.85, not 0.15.
💡 Top tips
Ask first: is a fixed amount changing it (arithmetic) or a fixed factor (geometric)?
Simple interest is arithmetic; compound interest is geometric — learn this pairing.
Turn a percentage into a ratio: +x% → 1 + x/100, −x% → 1 − x/100.
“On the nth day” wants a term; “total after n days” wants a sum.
If solving for n gives a decimal, test the whole numbers on either side.
âš Common mistakes
Choosing the wrong model — check whether the change is added or multiplied.
Using the decay percentage as the ratio — a 15% loss gives r = 0.85, not 0.15.
Confusing a term with a total — read whether one value or a cumulative sum is wanted.
Treating compound interest as arithmetic — it multiplies each period, so it is geometric.
Giving an unrealistic answer — a fraction of a person or a negative mass signals an error.
That completes Sequences & Series. You now have the full toolkit — the language of terms and sums, sigma notation, arithmetic and geometric models, and the real-world applications that tie them together. The deciding question is always the same: is the change a fixed amount, or a fixed factor?
Need help with AI HL Sequences & Series?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
