Find a recurrence relation and initial condition for...?

Question

Find a recurrence relation and initial condition for the number of fruit flies in a jar if there are 12 flies initially and every week there are six times as many flies in the jar as there were the previous week.

Answer 1

Let x_n be number of fruit files at the beginning of week n. Then x_0 = 12, which is the initial condition. And, for every n =1,2,3....

x_n = 6 x_(n-1) which is the recurrence relation.

From this, it follows that x_n = 12 * 6^n n =0, 1,2...

Answer 2

in case you do no longer prefer to apply or do no longer understand approximately producing purposes, one thank you to sparkling up the 2nd could be like this: If a(n) = - a(n-one million) +(n-one million), then a(n-one million) = - a(n-2) + (n-2) so, substituting, we get: a(n) = a(n-2) + one million From this, for even n, we get: a(n) = n/2 + a(0) (I advise you practice it by induction) For abnormal n, replace the above in the unique recurrence: a(n) = -a(n-one million) + (n-one million) = = -(n-one million)/2 + 7 + n-one million = = (n-one million)/2 + 7