Question

For the given statement Pn, write the statements P1, Pk, and Pk+1. (2 points) 2 + 4 + 6 + . . . + 2n = n(n+1)

Answer 1

P(1) = 2, P(k) = 2k, P(k+1) = 2(k+1)

Explanation:

The given statement is Pn = 2 + 4 + 6 +........+ 2n = n(n+1)

Now we have to write the statements P1, Pk, P(k+1)

P1 = 2

Pk = P1 + (n -1 ) d

Pk = 2 + (k -1)2 = 2 + 2k -2 = 2k

P(k+1) = Pk + 2

P(k+1) = 2k + 2 = 2(k+1)

Therefore the answers are

P1 = 2

P(k) = 2k

and P(k+1) = 2(k +1)

Answer 2

P n ( given statement ) : 2 + 4 + 6 + ...+ 2 n = n · ( n + 1 )
P 1 : 2 = 1 · ( 1 + 1 )
2 = 2
P k : 2 + 4 + 6 + ... + 2 k = k · ( k + 1 )
P k+1 :
2 + 4 + 6 + ... + 2 k + 2 · ( k + 1 ) = k · ( k + 1 ) + 2 · ( k + 1 ) =
= ( k + 1 ) · ( k + 2 )