Question

(3m5+6m4-8n3+5) + (3m4+4n3-9)​

Answer 1

Step-by-step explanation:

Let P(n):1+5+9+ . . .+(4n-3)=n(2n-1), for all natural number n.

Step I We observe that P(1) is true.

P(1):1=1(2×1−1),1=2−1P(1):1=1(2×1-1),1=2-1 and 1=1, which is true.

Step II Now assume that P(n) is true for n=k.

So, P(k):1+5+9+ . . .+(4k-3) = k(2k-1) is true.

Step III Now, to prove P(k+1) is true.

(P(k+1):1+5+9+. . . +(4k-3)+4k+1)-3

=k(2k-1)+4(k+1)-3

=2k2−k+4k+4−3=2k2-k+4k+4-3

=2k2+3k+1=2k2+3k+1

=2k2+2k+k+1=2k2+2k+k+1

2K(k+1)+1(k+1)

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

=(k+1)[2k+1+1-1]

=(k+1)[2(k+1)-1]

So, P(k+1) is true, whenever p(k) is true, hence p(n) is true.