# Geometric series

> Sums that shrink by the same factor: seed ÷ (1 − k) for referrals, LTV ≈ margin ÷ churn and lifetime 1 ÷ churn, the finite sum for annuities, capped LTV and a savings pot.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Growth and time
- Lesson 23/36, ~30 min
- Source: https://productovi.com/courses/math-basics/geometric-series

## Each step a share of the last

**Alex Tutor:** Still on Klasna's product team. The referral programme has data: on average each new family brings 0.3 more. Marketing plans 100 families from ads this month. How many arrive in the end?

**Sofia:** 100 from ads, then 30 from them, then 9 from those, then 2.7… It keeps going, but smaller each time.

**Alex Tutor:** That's a **geometric series**: each term is the previous one times the same factor q. When |q| < 1, the endless sum has a finite total.

$$
a + aq + aq^2 + aq^3 + \dots = \frac{a}{1 - q}, \quad |q| < 1
$$

*a: the first term; q: the factor from one term to the next.*

**Alex Tutor:** First, when does such a sum stop growing?

> **Exercise:** The endless sum 1 + 1.1 + 1.1² + 1.1³ + … adds up to a finite total.

**Alex Tutor:** With a = 100 and q = k = 0.3: 100 ÷ 0.7 ≈ 142.9 families. The referrals add seed × k ÷ (1 − k) = 100 × 0.3 ÷ 0.7 ≈ 42.9 on top.

![100 families from ads bring 30, who bring 9, then 2.7 and 0.81; the generations add up towards 100 ÷ (1 − 0.3) ≈ 142.9.](https://productovi.com/courses/math-basics/diagrams/m6-l3-referral-generations.png)

*Each generation is 0.3 of the one before.*

> **Exercise:** Ads bring 200 families and each new family brings k = 0.2 more on average. How many families arrive in total?

## Lifetime and LTV

**Sofia:** And if k were 1 or more?

**Alex Tutor:** Then the terms stop shrinking and the sum never settles: self-sustaining growth. Real programmes almost never get there, so plan with k well below 1.

**Alex Tutor:** The same sum hides in lifetime value. A Klasna Club member leaves 200 UAH of margin a month, and 5% leave each month, so each month 0.95 of members stay.

**Sofia:** Expected months paid: 1 + 0.95 + 0.95² + … = 1 ÷ (1 − 0.95) = 20. So the lifetime is 1 ÷ churn, and LTV ≈ 200 × 20 = 4,000.

$$
T \approx \frac{1}{c} \qquad LTV \approx \frac{m}{c}
$$

*T: expected lifetime in months; c: monthly churn; m: margin per month. Assumes churn stays the same every month and ignores discounting.*

> **Exercise:** A member leaves 150 UAH of margin a month and monthly churn is 4%. What is LTV ≈ m ÷ c, in UAH?

**Alex Tutor:** See how sensitive that lifetime is.

> **Exercise:** Monthly churn in Klasna Club falls from 5% to 4%. By how many months does the expected lifetime 1 ÷ c grow?

## When the sum stops

**Alex Tutor:** Often you only want the first n terms: a 12-month horizon, a five-year contract. The finite sum is the infinite one minus its tail.

$$
a + aq + \dots + aq^{n-1} = a \cdot \frac{1 - q^n}{1 - q}
$$

*n terms, from a to aqⁿ⁻¹. In a spreadsheet it is also just the sum of a column.*

**Sofia:** For the Club over 12 months: 200 × (1 − 0.95¹²) ÷ 0.05 ≈ 200 × 9.19 ≈ 1,839. Less than half of 4,000.

**Alex Tutor:** 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.

![With 5% monthly churn the expected months paid add up to 9.19 within 12 months and approach 20 forever, so a 200 UAH member is worth 1,839 within a year and about 4,000 in total.](https://productovi.com/courses/math-basics/diagrams/m6-l3-capped-ltv.png)

*The first year holds less than half of the endless sum.*

> **Exercise:** A member leaves 150 UAH of margin a month with 4% monthly churn. What is the 12-month capped LTV, 150 × (1 − 0.96¹²) ÷ 0.04, in UAH?

**Alex Tutor:** Discounting has the same shape. At 25% a year, money each year is worth 0.8 of the year before, because 1 ÷ 1.25 = 0.8.

**Sofia:** So five yearly payments of 1 are worth 0.8 + 0.64 + 0.512 + 0.4096 + 0.32768 today?

**Alex Tutor:** Yes: 2.68928, the **annuity factor**. With the formula, a = 0.8, q = 0.8, n = 5: 0.8 × (1 − 0.8⁵) ÷ 0.2 = 2.68928. Multiply by the payment and you have the present value of a loan or a contract.

**Sofia:** And a spreadsheet would just add the five discounted cells.

> **Exercise:** A bank loan is repaid with five yearly payments of 100,000 UAH, the first in one year. What is their present value at 25% a year?

## A pot that grows

**Alex Tutor:** Run the factor above 1 and you get a savings pot. Put P in every year at rate r: the first payment grows n − 1 times, the last not at all.

$$
S = P \cdot \frac{(1 + r)^n - 1}{r}
$$

*S: the pot after n years; P: the yearly payment, paid at the end of each year; r: the yearly rate; n: years. The same finite sum with a = P and q = 1 + r.*

**Sofia:** Three years of 10,000 at 10%: 12,100 + 11,000 + 10,000 = 33,100. And 10,000 × (1.331 − 1) ÷ 0.1 is also 33,100.

**Alex Tutor:** That's how the personal-finance course projects a pension pot. The formula needs a steady payment and a steady rate; real returns swing, so read the result as a range, not a promise.

**Sofia:** And to compare the pot with today's money, it still has to be brought back to today.

![A cat walks along a shelf of food bowls that get smaller from left to right, each holding less kibble.](https://productovi.com/courses/math-basics/inline/m6-l3-i1.webp)

*Each step is a share of the one before, so the total stays finite.*

> **Exercise:** You save 24,000 UAH at the end of every year for 20 years at 8% a year. Which expression gives the pot?

**Alex Tutor:** Before any of these numbers goes into a plan, check what it assumes.

> **Exercise:** LTV ≈ m ÷ c gives 4,000 UAH for a Club member with 200 UAH margin and 5% monthly churn. Which assumptions does that number rely on? Select all that apply.

## Your turn: what a member is worth

**Alex Tutor:** Leo wants to raise the Club's acquisition budget: "LTV is 200 ÷ 0.05 = 4,000. We can pay up to 4,000 per member and still break even."

**Sofia:** The formula is right. What it assumes isn't.

**Alex Tutor:** Then tell him what 4,000 assumes and what you'd use instead to set a budget.

> **Exercise:** Reply to Leo, Klasna's CEO, in 3–4 sentences: why is LTV = 200 ÷ 0.05 = 4,000 risky as an upper limit for acquisition cost, and what number would you use? Use the 12-month sum.

## Where you will meet this

> **In short**
>
> - **Infinite sum:** a + aq + aq² + … = a ÷ (1 − q), only for |q| < 1.
> - Referrals: total = seed ÷ (1 − k); added by referrals = seed × k ÷ (1 − k).
> - **Lifetime ≈ 1 ÷ churn**, **LTV ≈ margin ÷ churn**, assuming constant churn and no discounting.
> - **Finite sum:** a × (1 − qⁿ) ÷ (1 − q). It gives a capped LTV, the annuity factor (0.8 + … + 0.32768 = 2.68928 at 25%) and, with q = 1 + r, a savings pot P × ((1 + r)ⁿ − 1) ÷ r.

> **These lessons use it**
>
> - [Invites and viral growth](/courses/product-analytics/virality): total = seed ÷ (1 − k).
> - [Channels and marketing attribution](/courses/product-analytics/channels): break-even CAC with k ÷ (1 − k).
> - [The LTV curve](/courses/product-analytics/ltv-curve): a capped LTV read from a curve.
> - [Discounted LTV and CAC payback in cash](/courses/finance-for-product/ltv-and-cac-payback-in-cash): a finite, discounted sum.
> - [Case: build our own video classroom or keep paying per minute?](/courses/finance-for-product/build-or-buy-case): the factors 0.8, 0.64, … 0.32768.
> - [Raising the price](/courses/bootstrapped-product/raising-the-price): price ÷ churn as the value of a customer.
> - [Retirement: what the state gives and what you add](/courses/personal-finance/retirement): pot = P × ((1 + r)ⁿ − 1) ÷ r.

**Alex Tutor:** Next time: e and ln, and how they turn powers and odds into something you can solve.

- Previous lesson: [Discounting](https://productovi.com/courses/math-basics/discounting.md)
- Next lesson: [e, ln, odds and the sigmoid](https://productovi.com/courses/math-basics/e-ln-odds-sigmoid.md)

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