Prove formulas for every finite payment count using a base case and a general inductive step, and distinguish induction from numerical checks and infinite limits.
Mathematical induction
Mathematical induction is a proof method for statements indexed by integers: prove a starting case, then prove that each case implies the next. In quantitative work, it can certify a formula for every finite payment count or establish a property preserved by every iteration of an algorithm. Checking a few outputs can reveal a mistake; induction can establish the general claim.
The two obligations
Let P(n) be the statement to prove for every integer n≥n0. An induction proof establishes:
Base case:P(n0) is true.
Inductive step: for an arbitrary integer n≥n0, assume P(n) and deduce P(n+1).
These establish P(n) for all integers n≥n0. The starting value might be zero, one or another integer; choose it to match the claim. The logical quantifier “for every” in the second obligation matters: proving only the transition from four to five leaves the other transitions unproved. The University of Toronto’s induction guide sets out this base-case/step structure.
Assuming P(n) inside the step is legitimate because the immediate task is to prove a conditional implication. Assuming P(n+1), the conclusion of that implication, would leave the task undone.
A discounted sum, proved once
Consider a hypothetical stream paying one monetary unit at dates 0,1,…,n−1, with a stipulated discount factor of 1/2 per period. The date-zero payment is undiscounted. This deliberately simple factor is an arithmetic example, not an observed market rate.
Base case, n=1. The single payment is worth S1=1, and the proposed expression gives 2(1−1/2)=1.
Inductive step. Assume Sn=2(1−2−n) for an arbitrary integer n≥1. Adding the next payment gives
Sn+1=Sn+2−n=2(1−2−n)+2−n=2−2−n=2(1−2−(n+1)).
The second line is exactly where the induction hypothesis enters. The last line is the required formula for n+1. Together with the base case, this proves the identity for every positive integer payment count. □
For four payments, 1+0.5+0.25+0.125=1.875 monetary units. A fifth adds 0.0625, giving 1.9375. Those are useful arithmetic checks, while the arbitrary-n argument supplies the proof.
A base case P(1) and a general implication from P(n) to P(n+1) establish every positive integer case. Five finite geometric sums are 1, 1.5, 1.75, 1.875 and 1.9375 present-value units; the five checks alone do not prove the identity.
Original exact arithmetic for the hypothetical discount factor 1/2. The chain describes the proof obligations; the bars illustrate only five finite cases.
The update Sn+1=Sn+2−n is a recurrence relation. Induction verifies that the proposed closed form satisfies both that update and its starting value. This is useful when replacing a loop over payments with a direct expression: the proof connects the two mathematical calculations, although software still needs correct indexing and numerical arithmetic.
What the proof does not give you
A step without a base proves nothing about whether the chain starts. The false identity
1+2+⋯+n=2n(n+1)+1
survives the formal “add n+1 to both sides” manipulation. Yet at n=1 its two sides are 1 and 2. The failed base case prevents an induction proof.
All finite counts do not include an infinite count. For a general real ratio q=1, the finite identity is ∑k=0n−1qk=(1−qn)/(1−q). Replacing it with an infinite sum requires a separate limit argument. For ∣q∣<1, qn→0 and the sum tends to 1/(1−q); for q=2, the finite identity remains valid but the infinite series diverges. See the University of Waterloo’s finite and infinite geometric-series discussion.
Some steps need more than the immediately preceding case. Strong induction assumes all cases from the starting index through n when proving n+1. It is convenient for recurrences involving several earlier terms. It has the same logical power as ordinary induction: apply ordinary induction to the accumulated statement that all cases up to n hold. MIT’s induction chapter, especially §5.3 explains this equivalence.
For the separate question of when partial sums converge, SelfMadeQuant’s Series and the Geometric Series: Partial Sums and Convergence develops the limit argument and financial applications. The public page is a preview; the full lesson requires membership.
