数列

This resource is part of the 2025 archive of the Oxford MAT Livestream. Click or tap here or on the image below for the new and updated livestream admissions support programme from Oxford Mathematics. Sequences defined iteratively and by formulae. Arithmetic and geometric progressions . Their sums . Convergence condition for infinite geometric progressions . Hopefully this will give us some equations involving a and b. We're not too worried about finding all possible solutions here; we're just looking for anything that works, and that has a and b positive. Alternatively, try large numbers Y until you find one with 2Y^2+1 equal to a square number. This might take a while! It's a good idea to write out your work clearly, so that you have three equations in a tidy format, ready for the next part. For all of this to work, we'll need to see matching expressions for those differences. (iv) For t n, expand each bracket. Collect terms together, and watch out for a sum that we've already done. Consider the terms corresponding to A and B and C separately. Those are each just constants (which you know the value for!) and they can be brought outside each sum. For example, sum {k=1}^n left(A k 2^k right)=A sum {k=1}^n left(k 2^k right). Once again, you'll need to recognise a sum that you've already done.