if i could get a thurough explination please and thank youi :D Consider the sequ

Question

if i could get a thurough explination please and thank youi :D

Consider the sequence where a1 = 1 and an = an-1 + 3/5. The sequence is bounded below (You don't need to show this). Does the sequence have a limit? Justify. Find the limit

Explanation / Answer

Trivially, it is bounded below by 0 (we weren't asked to show this, but I wanted to show you; As an > 0, an + 3 > 0, and (an+3)/5 is> 0. Yes, it has a limit. We can easily solve for the limit. x = (x+3)/5 Cross-multiplying, 5x = x+3 4x = 3 x = 3/4 Let's prove this is the limit. Method 1. Show that all an > 3/4. Show that an 3/4, so write it as 3/4 + epsilon, epsilon > 0. Then, (3/4 + epsilon + 3)/5 = (3 3/4 + epsilon)/5 = 3/4 + epsilon/5 As epsilon > 0, 3/4 + epsilon/5 > 3/4 3/4 + epsilon/5 < 3/4 + epsilon/5 + 4epsilon/5 = 3/4 + epsilon = an-1, so an-1 infinity 3/4 + 5/4 * 1/5^n = 3/4

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