Grades 9-10 Video Solutions 2025-2025_9-10

Video Transcription

Problem 26, a sequence of numbers a1 through an is such that from the third term on each term is equal to the mean of all the previous terms.  So a3 is the mean of a1 and a2, a4 is the mean of a1, a2 and a3 and so on. In this sequence a1 is 8 and a10 is 26. We need to find the value of a2.  So we know that we can write all the terms in terms of a1 and a2. So we know that a3 is a1 plus a2 divided by 2.  Then a4 is a1 plus a2 plus a3 divided by 3, but we've already written a3 in terms of a1 and a2 so we can substitute that in and that simplifies to a1 plus a2 divided by 2 again.  Okay so this seems like a pattern. We know that a3 and a4 are both a1 plus a2 divided by 2. Does this mean that for all the other terms from a3 to a10 they're all going to be equal  to this? It turns out they are and we can prove it. So if we know that a3 through an minus 1 are a1 plus a2 divided by 2 for some n, it doesn't  matter, then we can prove that the next one, an, is also going to be equal to a1 plus a2 divided by n just by doing out the algebra.  So we know that an is the average of the first n minus 1 terms and we know that the first n minus 1 terms are a1, a2, and then n minus 3 copies of a1 plus a2 divided by 2 because  of the assumption we made. And then when we add up a1, a2, and n minus 3 copies of a1 plus a2 divided by 2, this becomes n minus 1 times a1 plus a2 divided by 2.  And so the an-th term is also a1 plus a2 divided by 2. So now we know that a3 through a10 are all the average of the first two terms.  This means that since a10 is 26 and a1 is 8, 26 is 8 plus a2 divided by 2. And we can solve this to get that a2 is 44. And the answer is D.

Video Summary

The sequence starts with \( a_1 = 8 \) and is defined such that each term from the third onward is the mean of all previous terms. It is revealed that \( a_3 \) through \( a_{10} \) are all equal to the average of \( a_1 \) and \( a_2 \). Given \( a_{10} = 26 \), the equation \( 26 = (8 + a_2) / 2 \) is derived. Solving this equation gives \( a_2 = 44 \). The final answer is \( D \).