For the given statement Pn, write the statements P1, Pk, and Pk+1.
2 + 4 + 6 + . . . + 2n = n(n+1)
Would the answer just be: (k+1)(k+2) ?
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well that is the sum of k + 1 terms....
the left side contains the terms including the general term 2n
the right side contains the sum of the terms n(n + 1)
so n = 1 term 1 = 2 the sum of 1 term 1(1 + 1) 2
the kth term 2k the sum of k terms is k(k + 1)
the k + 1 term 2(k + 1) the sum is (k +1)(k + 2)
this seems a lot like mathematical induction.
Yes, it is. But what should the final answer even look like? I'm confused by the problem.
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ok... so its true for n = 1
assume that for n = k the sum is k(k +1)
now for the k + 1 term
term k+ 1 = 2(k + 1) or 2k+ 2
if you add this to the sum of k terms you should get (k + 1)(k + 2)
so sum of term k + 1 and the sum of k terms
2(k + 1) + k(k + 1)
both terms have a common factor of (k + 1)
so it can be written as (k + 1)(2 + k) or (k + 1)(k + 2) (1)
now using the sum n(n + 1)
for k + 1 terms it becomes (k + 1)(k + 1+1) or (k + 1)(k+2)
so you have shown that the sum of k terms and the k + 1 term is equal to the sum of k +1 terms..
hope that makes sense