Question

For the given statement Pn, write the statements P1, Pk, and Pk+1.
(2 points)

2 + 4 + 6 + . . . + 2n = n(n+1)

Would the answer just be: (k+1)(k+2) ?

Answer 1

well that is the sum of k + 1 terms....

Answer 2

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.

Answer 3

Yes, it is. But what should the final answer even look like? I'm confused by the problem.

Answer 4

@campbell_st

Answer 5

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

Answer 6

So what I wrote is correct then?

Answer 7

@campbell_st

Answer 8

yes...

Answer 9

Ok thanks so much!