Any tips on how to conjecture formulas for Induction?

Click For Summary
Conjecturing formulas for mathematical induction often involves clever techniques rather than a fixed procedure. Reversing the order of sums and adding them can help in deriving formulas. For geometric series, multiplying the partial sum by the common ratio and using the equation S - rS can yield useful results. While generating functions can provide insights, they require a deeper understanding and patience to explore. Overall, creativity and familiarity with various mathematical tricks are essential in this process.
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.
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
View full post »

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
2K
Two Conjectures about Permutations in ##S_n##
  • · Replies 3 ·
Replies
3
Views
2K
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
1K
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
Proof by Induction of String exponentiation? (Algorithms)
  • · Replies 5 ·
Replies
5
Views
2K