Math Problem Statement

find the formula in this recurrence problem simple

find the formula in this recurrence problem simple

Solution

The recurrence relation in the image is as follows:

T(0)=1T(0) = 1 T(n)=T(n1)3+4T(n) = T(n-1) \cdot 3 + 4

The task is to find the explicit formula for T(n)T(n).


Step 1: Write out a few terms of the sequence

We already see the first few terms in the image: T(0)=1, T(1)=7, T(2)=25, T(3)=79, T(4)=241, T(5)=727T(0) = 1,\ T(1) = 7,\ T(2) = 25,\ T(3) = 79,\ T(4) = 241,\ T(5) = 727


Step 2: Analyze the recurrence relation

The recurrence can be rewritten as: T(n)4=3(T(n1)4)T(n) - 4 = 3 \cdot (T(n-1) - 4)

Let: S(n)=T(n)4S(n) = T(n) - 4

This transforms the recurrence into: S(n)=3S(n1)S(n) = 3 \cdot S(n-1)

Thus, S(n)S(n) is a geometric sequence with: S(0)=T(0)4=14=3S(0) = T(0) - 4 = 1 - 4 = -3 S(n)=33nS(n) = -3 \cdot 3^n


Step 3: Substitute back for T(n)T(n)

Since T(n)=S(n)+4T(n) = S(n) + 4, we have: T(n)=33n+4T(n) = -3 \cdot 3^n + 4


Final Explicit Formula:

T(n)=33n+4T(n) = -3 \cdot 3^n + 4


Verification:

  1. T(0)=330+4=3+4=1T(0) = -3 \cdot 3^0 + 4 = -3 + 4 = 1
  2. T(1)=331+4=9+4=7T(1) = -3 \cdot 3^1 + 4 = -9 + 4 = 7
  3. T(2)=332+4=27+4=25T(2) = -3 \cdot 3^2 + 4 = -27 + 4 = 25
  4. T(3)=333+4=81+4=79T(3) = -3 \cdot 3^3 + 4 = -81 + 4 = 79
  5. T(4)=334+4=243+4=241T(4) = -3 \cdot 3^4 + 4 = -243 + 4 = 241

Do you want to explore this recurrence further or have any questions? Here are some related questions:

  1. Can you explain how this sequence grows exponentially and why the 3n3^n term dominates?
  2. How would the explicit formula change if the multiplier or additive constant in the recurrence were different?
  3. Can we use this formula to compute terms for very large nn efficiently?
  4. What happens if the recurrence is modified to T(n)=T(n1)3+cT(n) = T(n-1) \cdot 3 + c (arbitrary cc)?
  5. How can we derive such explicit formulas for other types of recurrence relations?

Tip: Transforming a recurrence into a simpler form (like a geometric sequence) often makes it much easier to find an explicit solution.

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Math Problem Analysis

Mathematical Concepts

Recurrence Relations
Geometric Sequences
Algebra

Formulas

T(n) = -3 * 3^n + 4

Theorems

Geometric Sequence Formula

Suitable Grade Level

Grades 10-12