Please help thank you so much Consider the statement For all

Please help thank you so much!

Consider the statement: For all positive integers n, 1 + 5 + 9 +... + (4n - 3) = 2n^2 - n Identify P(n) and write down the statements which correspond to: P(1), P(4), P(k), P(k + 1) and P (2k - 1). (b) Prove by induction: n N 1 + 5 + 9 +... + (4n - 3) = 2n^2 - n

Solution

(a) P(n):1+5+9+...+(4n-3)=2n²-n

P(1)=1

P(4)= 1+5+9+13=28

P(k):1+5+9+...+(4k-3)=2k²-k

P(k+1): 1+5+9+...+(4k+1)=2(k+1)²-(k+1)

P(2k-1): 1+5+9+...+(8k-7)=2(2k-1)²-(2k-1)

(b) when n=1, LHS =1 AND RHS= 1.

Therefore the statement is true for n=1.

Let us assume the statement is true when n= k.

1+5+9+...+(4k-3)=2k²-k ----(1)

Now, we shall prove the statement when n = k+1

1+5+9+...+(4k-3)+(4k+1)=2k²-k+(4k+1) (by adding 4k+1 on both sides of (1))

=2k²+4k+2-k-1 (by re-writing the terms)

=2(k+1)²-(k+1)

Therefore the statement is true for n=k+1.

By Mathematical Induction, the given statement is true for all values of n(natural Numbers)