Any tips on how to conjecture formulas for Induction?

Click For Summary
SUMMARY

This discussion focuses on strategies for conjecturing formulas for mathematical induction, particularly in the context of summing sequences. Key techniques include reversing the order of terms and grouping them, as well as utilizing properties of geometric series by manipulating partial sums with the common ratio "r". The concept of generating functionology is highlighted as a valuable resource for deriving formulas, although it may require a deeper commitment to study. Overall, the discussion emphasizes the importance of clever tricks and individual problem-solving approaches in this area.

PREREQUISITES
  • Understanding of mathematical induction principles
  • Familiarity with geometric series and their properties
  • Basic knowledge of sequence summation techniques
  • Introduction to generating functions and their applications
NEXT STEPS
  • Study the properties of geometric series in depth
  • Explore generating functionology and its applications in combinatorics
  • Practice deriving formulas for various sequences using mathematical induction
  • Investigate advanced techniques in summation and series manipulation
USEFUL FOR

Mathematics students, educators, and anyone interested in enhancing their skills in mathematical induction and sequence analysis.

phillyolly
Messages
157
Reaction score
0
What are usual tips in conjecturing formulas for math induction if I am given a certain sum of sequence? Thank you.
 
Physics news on Phys.org
It depends on the problem.

In the case of sums...

Sometimes simply reversing the order and adding it to itself and then grouping terms is sufficient.

In the case of the geometric series simply multipliying the partial sum by the "r" of the geometric series and then performing S-rS can do some magic.

Usually there is some sort or clever trick involved and no one can really give you a procedure for coming up with clever tricks.

I only "general" way of coming up with formulas of that sort is generatingfunctionology ,yes it is actually a topic. I remember coming across an online book of that sort when I wanted to find a solution to a recursive integral. I didn't have the patience to read through more than 5pages though.
 

Similar threads

Prove this inequality using induction
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Proving Goldbach's conjecture by induction (or why it doesn't seem to work)
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Questions regarding Kurepa's Conjecture
  • · Replies 1 ·
Replies
1
Views
3K
Two Conjectures about Permutations in ##S_n##
  • · Replies 3 ·
Replies
3
Views
2K
Graduate About a natural extension of Collatz conjecture
  • · Replies 3 ·
Replies
3
Views
1K
Need help checking my answers on derivatives and proof by induction
  • · Replies 14 ·
Replies
14
Views
2K
Engineering Inductance of a layered Solenoid?
  • · Replies 2 ·
Replies
2
Views
2K
Induction with binomial coefficient
  • · Replies 15 ·
Replies
15
Views
3K
Using Euler's formula to prove trig identities using "sum to product" technique
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Swampland Conjecture: String Theory & Unstable Universes
  • · Replies 1 ·
Replies
1
Views
2K