UNDERSTANDING THE ENTIRE UNIVERSE AND MORE ABOUT SCIENCE

A recurrence relation is an equation that defines a sequence recursively, meaning each term is defined in terms of previous terms. Solving a recurrence means finding a closed-form solution that expresses the general term explicitly.

There are different types of recurrence relations, and various methods exist for solving them.

Types of Recurrence Relations
Linear vs. Nonlinear

Linear: Each term is a linear function of previous terms (e.g.,
𝑇
(
𝑛
)
=
3
𝑇
(
𝑛
−
1
)
+
2
T(n)=3T(n−1)+2).
Nonlinear: Involves products or nonlinear functions of previous terms (e.g.,
𝑇
(
𝑛
)
=
𝑇
(
𝑛
−
1
)
𝑇
(
𝑛
−
2
)
T(n)=T(n−1)T(n−2)).
Homogeneous vs. Non-Homogeneous

Homogeneous: No additional function is present (e.g.,
𝑇
(
𝑛
)
=
3
𝑇
(
𝑛
−
1
)
−
2
𝑇
(
𝑛
−
2
)
T(n)=3T(n−1)−2T(n−2)).
Non-Homogeneous: Includes an extra function (e.g.,
𝑇
(
𝑛
)
=
3
𝑇
(
𝑛
−
1
)
+
2
𝑛
T(n)=3T(n−1)+2n).
First-Order vs. Higher-Order

First-order: Depends only on the previous term (e.g.,
𝑇
(
𝑛
)
=
2
𝑇
(
𝑛
−
1
)
+
1
T(n)=2T(n−1)+1).
Higher-order: Involves multiple previous terms (e.g.,
𝑇
(
𝑛
)
=
𝑇
(
𝑛
−
1
)
+
2
𝑇
(
𝑛
−
2
)
T(n)=T(n−1)+2T(n−2)).
Divide and Conquer Recurrences

Common in algorithms like Merge Sort (e.g.,
𝑇
(
𝑛
)
=
2
𝑇
(
𝑛
/
2
)
+
𝑛
T(n)=2T(n/2)+n).
Methods for Solving Recurrence Relations
1. Iteration (Substitution) Method
Expand the recurrence multiple times to identify a pattern.
Useful for simple recurrence relations.
Often combined with induction for verification.
2. Recursion Tree Method
Expands the recurrence into a tree structure.
Useful for divide-and-conquer recurrences.
Helps analyze the total work done at each level.
3. Master Theorem
Provides a direct formula for solving divide-and-conquer recurrences of the form:
𝑇
(
𝑛
)
=
𝑎
𝑇
(
𝑛
/
𝑏
)
+
𝑓
(
𝑛
)
T(n)=aT(n/b)+f(n)
Cases compare
𝑓
(
𝑛
)
f(n) with
𝑛
log
⁡
𝑏
𝑎
n
log
b
​
a
to determine complexity.
Used in algorithm analysis, such as Merge Sort and Quick Sort.
4. Characteristic Equation Method
Applies to linear homogeneous recurrence relations.
Forms a characteristic equation whose roots determine the general solution.
Works well for relations like Fibonacci and constant-coefficient recurrences.
5. Generating Functions
Converts the recurrence into an algebraic equation in terms of a function.
Useful for complex recurrences where other methods are hard to apply.
Conclusion
Solving recurrence relations is essential in computer science, especially in algorithm analysis. The method chosen depends on the recurrence type, with the Master Theorem being particula A recurrence relation is an equation that defines a sequence recursively, meaning each term is defined in terms of previous terms. Solving a recurrence means finding a closed-form solution that expresses the general term explicitly.

There are different types of recurrence relations, and various methods exist for solving them.

Types of Recurrence Relations
Linear vs. Nonlinear

Linear: Each term i

1. UNDERSTANDING THE UNIVERSE AND THE GALAXY

2. My goal is simple. It is a complete understanding of the universe, why it is as it is and why it exists at all. — Stephen W. Hawking

3. What is the universe ? The universe is everything.it includes all of space and all the matter and energy that space contains.it even includes time itself and of course,it includes you.

4. Formation of the universe B 1.Big bang 2.Formation of the supernova and the nebula 3.Formation of the universe

5. When cosmic inflation stopped, the energy driving it transferred to matter and light – the big bang. One second after the big bang, the universe consisted of an extremely hot primordial soup of light and particles ( supernova ).

6. Supernova The supernova is colossal expansion of a star

7. After five minutes, most of today’s helium had formed ( nebula ) and the universe had expanded. In the following minutes, an era called nucleosynthesis, protons and neutrons collided and produced the earliest elements – hydrogen, helium, and traces of lithium and beryllium.

8. Nebula The Nebula is the giant cloud of dust and gas in space It comes from explosion of the supernova

9. Is the universe expanding ? Astronomer Edwin Hubble confirmed that the universe was expanding in 1929 using observations made by his associate, astronomer Milton Humason . Humason measured the redshift of spiral galaxies.

10. They compared the distances of these galaxies to their redshift and tracked how the farther away an object is, the bigger its redshift and the faster it is moving away from us– an observation now known as Hubble’s Law. The universe, they confirmed, is really expanding.

11. Galaxy What is the galaxy ? The galaxy is the huge collection of gas , dust and billions of stars and their solar system all held together by gravity What is the name of our galaxy?

12. Milky way galaxy The Milky way galaxy contains around 200 billion stars 3916 solar systems What is the name of neighbouring galaxy?

13. Andromeda galaxy

14. What is the distance between Milky way galaxy and Andromeda galaxy ? 2450 kly

15. Scientist words about the universe The eventual goal of science is to provide a single theory that describes the whole universe. — Stephen W. Hawking

16. A nice journey from the earth to the universe

17. THANK YOU Presented by Hari R N1-section Reg no:127003086