In this video I go over an example of a sequence that is defined by a recurrence relationship, and then use mathematical induction to prove it is increasing, bounded, thus convergent, and finally its actual limit. The first term of the sequence is 2, and the next terms are governed by the recurrence relation 1/2 (previous term + 6). Calculating these out manually suggests the terms are approaching the number 6. This suggests we use Mathematical induction by assuming the that terms are increasing and thus proving they are when starting with the initial term. I prove the sequence is bounded and increasing, hence the monotonic sequence theorem states it is convergent, but this doesn't tell us the value of the limit. I determine the limit using the recurrence relation in the limit as the terms go to infinity, and finally obtain the answer, which is the limit is 6. Amazing stuff!
#math #sequences #mathematicalinduction #calculus #education
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