Geometric series

A geometric series is a sum where each term is the previous one times the same factor q: a + aq + aq² + aq³ + … Here a is the first term and q the step factor.

When q is between −1 and 1, the terms shrink fast enough for the endless sum to settle on a finite total: a ÷ (1 − q). Why: call the sum S. Then qS is the same sum without its first term, so S − qS = a, and S = a ÷ (1 − q).

Example: 100 + 30 + 9 + 2.7 + … has a = 100 and q = 0.3, so the total is 100 ÷ 0.7 ≈ 142.9. The first three terms already give 139.

When q is 1 or more, the terms never shrink and the sum grows without limit. 1 + 1.1 + 1.21 + … has no total, and plugging it into a ÷ (1 − q) gives a meaningless negative number, 1 ÷ (−0.1) = −10. Always check |q| < 1 before you use the formula.

Referrals are a geometric series in disguise. If every new family brings k more on average, a seed of families from ads grows generation by generation: seed, seed × k, seed × k², …

Total families = seed ÷ (1 − k). Families added by referrals = seed × k ÷ (1 − k).

Example: Klasna's ads bring 100 families, and each new family brings 0.3 more. Total = 100 ÷ 0.7 ≈ 142.9; referrals add about 42.9 on top. The generations run 100, 30, 9, 2.7, 0.81…, so almost all of the effect comes in the first two or three.

Reading k: it's an average over everyone, including the many who bring nobody. If k reached 1, every family would replace itself and the chain would never stop: self-sustaining growth, which real programmes almost never reach. Plan with k well below 1.

The common slip is counting only the first generation, 100 + 30 = 130, or computing 100 × 1.3. The chain keeps going: the 30 bring 9 more.

Lifetime value hides the same series. Say a Klasna Club member leaves 200 UAH of margin a month and 5% of members leave each month. A member surely pays for month 1, is still there in month 2 with a chance of 0.95, in month 3 with 0.95², and so on.

Expected months paid = 1 + 0.95 + 0.95² + … = 1 ÷ (1 − 0.95) = 1 ÷ 0.05 = 20. In general, expected lifetime ≈ 1 ÷ churn and LTV ≈ margin ÷ churn: 200 ÷ 0.05 = 4,000 UAH.

The formula rests on two assumptions: churn is the same every month, and money in month 40 counts as much as money today, with no discounting as in Discounting. Real churn is usually higher in the first months and lower later, so treat m ÷ c as a quick estimate, not a fact.

Use margin, not revenue. An LTV built on revenue overstates what a member is worth, because tutors' pay and payment fees take part of every payment.

Expected lifetime 1 ÷ c isn't a straight line in churn: it gets steep as churn gets small. From 5% to 4% monthly churn the lifetime goes from 20 to 25 months; from 2% to 1%, from 50 to 100 months. Halving churn always doubles the lifetime, and so doubles LTV ≈ m ÷ c.

That's why retention work pays off more than it looks: a one-point drop in churn means a 25% longer life at 5%, and much more when churn is already low.

It's also why the formula is fragile. At low churn most of the value sits years ahead, exactly where "churn stays the same" is weakest. 1% monthly churn implies an average member stays over 8 years; few products have data that long.

So report a lifetime together with its churn and its horizon: "about 25 months at 4% monthly churn, if churn holds". When churn is tiny, prefer a capped sum over a horizon you trust to 1 ÷ c.

Often you need only the first n terms: a 12-month horizon, a five-year contract. The finite sum is a + aq + … + aqⁿ⁻¹ = a × (1 − qⁿ) ÷ (1 − q). It is the endless sum a ÷ (1 − q) minus its tail, which starts at aqⁿ and adds up to aqⁿ ÷ (1 − q).

For the Club member, a = 200 and q = 0.95 over 12 months: 0.95¹² ≈ 0.5404, so the expected months paid are (1 − 0.5404) ÷ 0.05 ≈ 9.19, and the 12-month value is 200 × 9.19 ≈ 1,839 UAH, less than half of the endless 4,000.

That's a capped LTV: the value within a horizon you trust. The endless sum keeps adding small amounts for years, and nobody knows whether churn will hold that long. For budgets, a 12- or 24-month capped LTV is the safer number.

Check by brute force: in a spreadsheet, a column of 200 × 0.95ᵏ for k = 0 to 11 adds up to the same 1,839. n counts terms, starting from k = 0.

Discounting has the same shape. At 25% a year, each year's money is worth 0.8 of the year before, because 1 ÷ 1.25 = 0.8. So five yearly payments of 1, the first in one year, are worth 0.8 + 0.64 + 0.512 + 0.4096 + 0.32768 = 2.68928 today. That sum is the annuity factor.

With the finite-sum formula: a = 0.8, q = 0.8, n = 5, so 0.8 × (1 − 0.8⁵) ÷ (1 − 0.8) = 0.8 × 0.67232 ÷ 0.2 = 2.68928. In general, for a rate r and n payments, the factor is (1 − (1 + r)⁻ⁿ) ÷ r.

Multiply by the payment to get the present value of a loan, a lease or a contract: five payments of 50,000 UAH at 25% are worth 50,000 × 2.68928 = 134,464 today, against 250,000 paid in total.

Check the timing. The factor assumes the first payment comes after one period. If the first payment is today, it isn't discounted, and the factor becomes 1 + 0.8 + 0.64 + 0.512 + 0.4096 = 3.3616.

Run the factor above 1 and the series becomes a savings pot. Pay P at the end of every year at rate r for n years: the last payment hasn't grown, the one before grew once, the first grew n − 1 times. Read from the last: P + P(1 + r) + … + P(1 + r)ⁿ⁻¹, a finite geometric series with a = P and q = 1 + r.

The sum is S = P × ((1 + r)ⁿ − 1) ÷ r. Check with three years of 10,000 at 10%: 12,100 + 11,000 + 10,000 = 33,100, and 10,000 × (1.331 − 1) ÷ 0.1 = 33,100.

Two shortcuts look plausible and are wrong: P × n × (1 + r) ignores compounding, and P × (1 + r)ⁿ grows only the first payment, as if nothing else had been paid in.

The formula needs a steady payment and a steady rate. Real returns swing, so read a projected pot as a range, not a promise. And the pot is in future money: to compare it with today, discount it back.

Every closed formula in this lesson is the sum of a series with fixed parts, so it inherits their assumptions. Before you put one into a plan, check four things:

• |q| < 1 for an endless sum; otherwise there is no total.
• The step factor is constant: the same churn every month, the same k in every referral generation, the same rate every year. Real churn falls after the first months; real returns swing.
• The amount per step is constant: the same margin, the same payment.
• Time is priced in: m ÷ c counts month 60 as fully as month 1; discounting would shrink it.

When an assumption is shaky, use a capped sum over a horizon you trust, or simply a spreadsheet column with the real numbers per period.

How it fits together: one shape, a + aq + aq² + …, gives referral totals, lifetime and LTV, capped LTV, annuity factors and savings pots. The next lesson adds e and ln, which find the exponent itself.