Diberikan bilangan asli 3,5,8,12,17,23,30,38, rancang suatu formula untuk mencari suku ke 400 dan lanjutkan dgn induksi matematika

Google translate - indonesian:

"Given the original numbers design a formula to find the 400th term and continue (prove) with mathematical induction"

Let  represent the n'th term of the sequence, that is

  is the first term,   the second term and so on...

notice that :

 

 

 

 

 

so clearly

 

Recall the famous Gauss formula for addition of the first consecutive n natural numbers: 1+2+3+...(n-1)+n=n(n+1)/2

then, the formula for u_n is:



for example, the 5th term is :



Proof by induction.

step 1
, true

step2

assume true for n=k, that is assume

step 3

verify for n=k+1, that is verify that .

because each nth term of the sequence is clearly its previous term +n

for example the fourth term 12, is the third term +4.

the fifth term, 17, is the fourth term 12 + 5 and so on...

also, by our assumption 

so:


factorizing k+1



which is what we needed to show.