This is an optional module for those preparing for Engineering examinations Q-1) Do there exist distinct positive integers such that form an arithmetic progression (in some order)? Q-2) Define a sequence by , and for . a) Show that for and , divides b) If divides , show that divides . Q-3) Consider a nonconstant arithmetic progression . Suppose there exist relatively… Continue reading
Tag: RMO
Week -7 | Sequences & Series – 7
This is an optional module for the students preparing for engineering entrance examinations Q-1) Show that tere cannot be an infinite AP, all of whose terms are perfect squares. Solution: Assuming that there exists such an AP, let denote the common difference of this progression. Thus can be represented as the difference of 2 perfect… Continue reading
Week -6 | Sequences & Series – 5
This is an optional module for the students preparing for engineering entrance examinations 1.1 Difference Equations By now that the difference between sequences and series is clear, we look at a new concept to solve recurrence relations related to sequences. Often we have our sequences defined by such a recurrence relation. For example, , with … Continue reading
Week -4 | System of Equations – 7
This is an optional module for the students preparing for engineering entrance examinations Q-1) Let be a positive integer such that is prime. Choose for , such that the are not all equal. Also let be a polynomial such that for all . Show that the degree of is at least . Q-2) Let be distinct integers.… Continue reading
Week -4 | System of Equations – 5
This is an optional module for the students preparing for engineering entrance examinations Q-1) Find the number of natural numbers which satisfy the following 2 conditions: a) b) divides Solution: , this is divisible by for any natural number 2. Note that for any number to be divisible by , it must be divisible by its… Continue reading