Let `C_k = .^nC_k,` for `0 lt= k lt=n` and `A_k = ((C_(k-1)^2,0), (0,C_k^2))` for `k gt= 1` and `A_1+A_2+...+A_n = ((k_1,0), (0,k_2)),` then

Let, C_(k) = ""^(n)C_(k) " for" 0 le kle n and A_(k) = [[C_(k-1)^(2),0],[0,C_(k)^(2)]] for k ge l and A_(1) + A_(2) + A_(3) +...+ A_(n) = [[k_(1),0],[0, k_(2)]] , then

Let, C_(k) = ""^(n)C_(k) " for" 0 le kle n and A_(k) = [[C_(k-1)^(2),0],[0,C_(k)^(2)]] for k ge l and A_(1) + A_(2) + A_(3) +...+ A_(n) = [[k_(1),0],[0, k_(2)]] , then

Let, C_(k) = ""^(n)C_(k) " for" 0 le kle n and A_(k) = [[C_(k-1)^(2),0],[0,C_(k)^(2)]] for k ge l and A_(1) + A_(2) + A_(3) +...+ A_(n) = [[k_(1),0],[0, k_(2)]] , then

Let, C_(k) = ""^(n)C_(k) " for" 0 le kle n and A_(k) = [[C_(k-1)^(2),0],[0,C_(k)^(2)]] for k ge l and A_(1) + A_(2) + A_(3) +...+ A_(n) = [[k_(1),0],[0, k_(2)]] , then

Let a_(k)=.^(n)C_(k) for 0<=k<=n and A_(k)=[[a_(k-1),00,a_(k)]] for 1<=k<=n and B=sum_(k=1)^(n-1)A_(k)*A_(k+1)=[[a,00,b]]

Find the sum sum_(k=0)^n ("^nC_k)/(k+1)

If a lt c lt b , and if 1 -k_(1) lt ln ((b)/(a)) lt k_(2) -1 , then ( k _(1) , k_(2)) is

Use the principle of mathematical induction : A sequence a_1, a_2, a_3,…… is defined by letting a_1 = 3 and a_k = 7a_(k-1) , for all natural numbers k > 2 . Show that a_n = 3.7^(n-1) , for all natural numbers .