Use mathematical induction to prove that 2n-1 (2i + 1) = 3n² i=n

The binomial theorem states that for any real numbers a and b, (a + b)" = -o ()a"-kbk for any integer n 2 0. Use this theorem to compute the coefficient of x when (2x 1) is expanded.

Show that C(2n, n)+C(2n, n−1)= 1/2C(2n+2, n+1).

Part C Please

Use induction to prove: for any integer n ≥ 1,(6-4)= 3n²-n. Base case n = Inductive step k+1 j=1 Assume that for any k > " = = j=1 (6j-4)=(6j- 4)+ j=1 =( (6j-4)= = 3 k² + k²+ k+ k+ j=1 j=1 , 3n²-n= Σ(6j-4)= )-(k+ 1) -(k+ 1) we will prove that By inductive hypothesis k+1 j=1 (6j-4)=

Use the method of proof by induction to show 2"+2 + 32n+1 is divisible by 7 Vn ε Zn 2 0.

n ≥ 0. Use simple induction to prove that ΣΙ 2 - n+1 = vi=1 n2n+2+2, for

Prove by induction:  gn=gn-1+gn-2+gn-4   for n greater than or equal to 4, with g0=1, g1=1, g2=2, g3=3, g4=6

Part D Please

1) Guess a formula for the sum 1-3+5-7+.. +(-1)*-(2n – 1) =>(-1)-(2i – 1) 1=1 and prove your formula is correct using induction