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Proof by induction question

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Reply 1
This is the question:
4FC31966-41E4-4D02-A8C3-37DF23287BF7.jpeg
This is the solution:
0DAEE86A-8B0E-47A8-B841-0295399D7C65.jpeg

I underlined the part of the solution that I didn’t understand. Why (2k+1)^2(2k+2)^2? Why not simply (2k+2)^2? Because 2(k+1) = 2k+2, which if you put into r^2, would be (2k+2)^2, but again that’s wrong
Original post by Sha.xo527

I underlined the part of the solution that I didn’t understand. Why (2k+1)^2(2k+2)^2? Why not simply (2k+2)^2? Because 2(k+1) = 2k+2, which if you put into r^2, would be (2k+2)^2, but again that’s wrong


It's because, when you change the upper limit of the summation from 2k to 2(k + 1), you're actually adding two terms to the sequence: r = 2k + 1 and r = 2k + 2.
Reply 3
Original post by old_engineer
It's because, when you change the upper limit of the summation from 2k to 2(k + 1), you're actually adding two terms to the sequence: r = 2k + 1 and r = 2k + 2.


Ah makes sense. Thank you

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