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Why Can We Add Solutions Together? Understanding the Principle of Superposition

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Last Updated on June 20, 2026 by Statnzee Team

Introduction

When studying recurrence relations such as those found in the Gambler’s Ruin problem, we often encounter a statement like:

If Aᵢ and Bᵢ are solutions, then any linear combination of them is also a solution.

At first glance, this may seem surprising. Why should adding two solutions produce another solution?

The answer lies in a fundamental property of linear equations known as the Principle of Superposition.

The Recurrence Relation

Suppose we are trying to solve the recurrence relation

P_i = pP_{i+1} + qP_{i-1}

where:

  • Pᵢ is the unknown sequence
  • p and q are constants
  • p+q=1

This is called a linear second-order recurrence relation.

Assume We Already Know Two Solutions

Suppose we have somehow found two sequences:

Aᵢ

and

Bᵢ

that satisfy the recurrence relation.

For Aᵢ we have:

A_i = pA_{i+1} + qA_{i-1}

For Bᵢ we have:

B_i = pB_{i+1} + qB_{i-1}

Since both equations satisfy the same recurrence, we call Aᵢ and Bᵢ solutions.

Multiplying a Solution by a Constant

Take the first solution and multiply every term by a constant C₁:

C_1A_i = p(C_1A_{i+1}) + q(C_1A_{i-1})

Notice that the equation remains valid.

The same is true for Bᵢ:

C_2B_i = p(C_2B_{i+1}) + q(C_2B_{i-1})

This tells us that multiplying a solution by a constant creates another solution.

Adding Two Solutions

Now add the two equations together:

Left side:

C_1A_i + C_2B_i

Right side:

p(C_1A_{i+1}+C_2B_{i+1}) + q(C_1A_{i-1}+C_2B_{i-1})

The resulting equation has exactly the same structure as the original recurrence relation.

Defining a New Sequence

Let

P_i = C_1A_i + C_2B_i

Then

P_{i+1}=C_1A_{i+1}+C_2B_{i+1}

and

P_{i-1}=C_1A_{i-1}+C_2B_{i-1}

Substituting these expressions into the previous equation gives

P_i = pP_{i+1}+qP_{i-1}

which is exactly the original recurrence relation.

Therefore Pᵢ is also a solution.

Why Does This Work?

The recurrence relation contains only:

  • Addition
  • Multiplication by constants

There are no powers, square roots, products of unknown terms, or other nonlinear operations.

Because of this, the equation preserves linear combinations.

This special property is called linearity.

A Simple Numerical Example

Suppose two solutions are

A_i = 1

and

B_i = \left(\frac{q}{p}\right)^i

Choose

C_1=3,\quad C_2=5

Then the new sequence becomes

P_i = 3 + 5\left(\frac{q}{p}\right)^i

Without any additional work, this new sequence is also a solution.

Why This Is Important

The Principle of Superposition allows us to build an entire family of solutions from a small set of basic solutions.

If we know two independent solutions:

1

and

\left(\frac{q}{p}\right)^i

then every solution can be written as

P_i = A + B\left(\frac{q}{p}\right)^i

where A and B are constants.

This expression is called the general solution.

Why Doesn’t This Work for Nonlinear Equations?

Consider the equation

y^2 = y

Two solutions are

y=0

and

y=1

If we add them together, we obtain

y=2

but

2^2 \neq 2

Therefore the sum is not a solution.

The superposition principle works only for linear equations, not nonlinear ones.

Key Takeaway

The Principle of Superposition states that:

Any linear combination of solutions to a linear recurrence relation is itself a solution.

This is why, after finding the two independent solutions

1

and

\left(\frac{q}{p}\right)^i

we can combine them to obtain the general solution

P_i = A + B\left(\frac{q}{p}\right)^i

This idea appears throughout mathematics, including recurrence relations, differential equations, linear algebra, probability theory, and physics.

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