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MathChapter 12: Composition, Recursion, and Exponential Functions

Notes

Recursive Formula Notes

Build recursive sequences from an initial condition and recurrence rule, track indexing, and model repeated change without restarting.

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A recursive formula defines a term using one or more preceding terms. It needs both a starting point and a rule for moving forward. The essential habit is to use the value just calculated, not the original value, at each new step.

Learning objectives

  • Identify the initial condition and recurrence equation.
  • Respect whether indexing begins at \(a_0\), \(a_1\), or another stated term.
  • Calculate every intermediate term in order.
  • Translate repeated percent change plus a fixed addition into a recurrence.

The two required parts

Initial condition versus recurrence rule

Initial condition

States where the sequence starts, such as \(a_0=6\) or \(a_1=6\).

Recurrence equation

Explains how to obtain a new term from a preceding term, such as \(a_n=2a_{n-1}+n\).

Each term depends on the term before itA four-term sequence begins at an explicit initial condition and repeatedly applies the same recurrence rule.
  1. \(a_0\)

    The stated starting condition.

  2. \(a_1\)

    Apply the rule to the initial value.

  3. \(a_2\)

    Apply the rule to the previous result.

  4. \(a_3\)

    Apply the rule again without restarting.

Calculate one term at a time

Fresh example recurrence
\[a_0=3,\qquad a_n=2a_{n-1}+n\quad(n\ge1)\]

The rule depends on both the previous term and the current index \(n\).

Recursive calculation from the previous term
IndexPrevious termRule appliedNew term
\(1\)\(a_0=3\)\(2(3)+1\)\(a_1=7\)
\(2\)\(a_1=7\)\(2(7)+2\)\(a_2=16\)
\(3\)\(a_2=16\)\(2(16)+3\)\(a_3=35\)

Repeated function composition

Worked example

Apply one function repeatedly

Let \(f(x)=2x-1\). Find \(f(f(f(3)))\).

  1. First application

    \(f(3)=2(3)-1=5\).

  2. Second application

    \(f(5)=2(5)-1=9\).

  3. Third application

    \(f(9)=2(9)-1=17\).

\(f(f(f(3)))=17\). Each result becomes the next input.

Recursive real-world models

Percentage change followed by a fixed contribution
\[A_n=(\text{growth or retention factor})A_{n-1}+\text{fixed contribution}\]

Follow the formula's operation order exactly for each period.

Worked example

A savings recurrence

An account begins with \(A_0=1200\). Each month it earns \(1\%\), then receives \(75\): \(A_n=1.01A_{n-1}+75\). Find \(A_2\).

  1. Month one

    \(A_1=1.01(1200)+75=1287\).

  2. Month two

    \(A_2=1.01(1287)+75=1374.87\).

  3. Order

    The previous balance is multiplied first; the contribution is added afterward.

After two months, the balance is \(1374.87\).

Common mistakes and traps

  • Using \(a_0\) at every step instead of the newly calculated term.
  • Treating a sequence beginning at \(a_0\) as though it begins at \(a_1\).
  • Jumping to a later term while skipping required intermediate values.
  • Ignoring the \(n\)-dependent part of the rule.
  • Adding a fixed contribution before multiplication when the recurrence specifies the reverse.
  • Rounding a decimal recurrence too early.
  • Confusing a recursive rule with an explicit formula that computes a term directly.
Mini check

Use the correct previous term

If \(b_1=4\) and \(b_n=3b_{n-1}-2\), what is \(b_3\)?

  1. \(10\)
  2. \(16\)
  3. \(28\)
  4. \(34\)
Show answer and explanation

Answer: \(28\)

\(b_2=3(4)-2=10\), then \(b_3=3(10)-2=28\).

Key takeaways

Key takeaways

What to remember

  • A recursive definition needs an initial condition and a recurrence rule.
  • The stated index determines which term comes next.
  • Each calculated term becomes the input for the following step.
  • In a mixed recurrence, preserve multiplication/addition order and delay rounding.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.