Prove that `sum_(r=0)^n 3^r n` `Cundersetr` = `4^n`.

Evaluate sum_(r=1)^(n)rxxr!

The natural numbers arearranged innthe form given below The rth group containing 2^(r-1) numbers. Prove that sum of the numbers in the nth group is 2^(n-2)[2^(n)+2^(n+1)-1] .

Prove by mathematical induction that sum_(r=0)^(n)r^(n)C_(r)=n.2^(n-1), forall n in N .

Let k=1^@ , then prove that sum_(n=0)^88 1/(cosnk* cos(n+1)k)=cosk/sin^2k

sum_(r=1)^n(2r+1)=...... .

If D_r=|2^(r-1)2(3^(r-1))4(5^(r-1))x y z2^n-1 3^n-1 5^n-1| then prove that sum_(r=1)^n D_r=0.

If alpha_(1), alpha_(2), alpha_(3), beta_(1), beta_(2), beta_(3) are the values of n for which sum_(r=0)^(n-1)x^(2r) is divisible by sum_(r=0)^(n-1)x^(r ) , then the triangle having vertices (alpha_(1), beta_(1)),(alpha_(2),beta_(2)) and (alpha_(3), beta_(3)) cannot be

sum_(r=0)^(n).^(n)C_(r)4^(r)=..........

Statement -1 If Delta(r)=|{:(r,r+1),(r+3,r+4):}| then sum_(r=1)^(n) Delta(r)=-3n Satement-2 If Delta(r)=|{:(f_(1)(r),f_(2)(r)),(f_(3)(r),f_(4)(r)):}| Sigma_(r=1)^(n) Delta (r)={:abs((Sigma_(r=1)^(n)f_1(r),Sigma_(r=1)^(n)f_2(r)),(Sigma_(r=1)^(n)f_3(r),Sigma_(r=1)^(n) f_4(r))):}

(i) Write sum_(r=1)^(n)(r^(2)+2) in expanded form. (ii) Write the series (1)/(3)+(2)/(4)+(3)/(5)+(4)/(6)+"…"+(n)/(n+2) in sigma form.