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Introduction to Mathmuni (www.mathmuni.com)
Introduction to Mathmuni (www.mathmuni.com)
Solve the equation x^4 - 4x^2 + 8x + 35 = 0 having one root as 2 + i*sqrt(3).
Solve the equation x^4 - 4x^2 + 8x + 35 = 0 having one root as 2 + i*sqrt(3).
Find the sum of the series 6 + 24 + 60 + 120 + 210 + 336 + ... to n terms.
Find the sum of the series 6 + 24 + 60 + 120 + 210 + 336 + ... to n terms.
Prove that a^2/c + b^2/d = 1, x = a/c, y = b/d if the following equations have only one solution.
Prove that a^2/c + b^2/d = 1, x = a/c, y = b/d if the following equations have only one solution.
Find the sum of the first n terms of the series whose rth term is (2r+1)/[r(r+1)]^2.
Find the sum of the first n terms of the series whose rth term is (2r+1)/[r(r+1)]^2.
Sum the series 1!/(a+1) + 2!/[(a+1)(a+2)] + ... + n!/[(a+1)(a+2) ... (a+n)].
Sum the series 1!/(a+1) + 2!/[(a+1)(a+2)] + ... + n!/[(a+1)(a+2) ... (a+n)].
Form an equation whose roots are the product of every two roots of x^3 - ax^2 + bx + c = 0.
Form an equation whose roots are the product of every two roots of x^3 - ax^2 + bx + c = 0.
Find nth term, sum to n terms and generating function of series 1+5x+7x^2+17x^3+31x^4+...
Find nth term, sum to n terms and generating function of series 1+5x+7x^2+17x^3+31x^4+...
Find an, bn in terms of a, b and show that when n is infinite an = bn.
Find an, bn in terms of a, b and show that when n is infinite an = bn.
Solve the equation sqrt(2x-1) + sqrt(3x-2) = sqrt(4x-3) + sqrt(5x-4).
Solve the equation sqrt(2x-1) + sqrt(3x-2) = sqrt(4x-3) + sqrt(5x-4).
Find the coefficient of x^n in expansion of (1+2x)/[1-x+x^2].
Find the coefficient of x^n in expansion of (1+2x)/[1-x+x^2].
Show that pq = r if two roots of x^3 + p*x^2 + qx + r = 0 be equal and of opposite sign.
Show that pq = r if two roots of x^3 + p*x^2 + qx + r = 0 be equal and of opposite sign.
Find 1 - k1 + k2 - ... if [1 + x + x^2]^(3r) = 1 + k1*x + k2*x^2 + ...
Find 1 - k1 + k2 - ... if [1 + x + x^2]^(3r) = 1 + k1*x + k2*x^2 + ...
Show that (a^2)[1+b^2] + (b^2)[1+c^2] + (c^2)[1+a^2] is greater than equal to 6abc.
Show that (a^2)[1+b^2] + (b^2)[1+c^2] + (c^2)[1+a^2] is greater than equal to 6abc.
If a^2+b^2 = 7ab, show that ln{(a+b)/3} = {lna + lnb}/2.
If a^2+b^2 = 7ab, show that ln{(a+b)/3} = {lna + lnb}/2.
Show that (A-C)(B-C)(A+D)(B+D) = q^2 - p^2 where A, B, C, D are defined as follows.
Show that (A-C)(B-C)(A+D)(B+D) = q^2 - p^2 where A, B, C, D are defined as follows.
Prove that (1-x)^(-3) = s1 + s2*x + s3*x^2 + ... + sn*x*(n-1) + ...
Prove that (1-x)^(-3) = s1 + s2*x + s3*x^2 + ... + sn*x*(n-1) + ...
Show that roots of (1+z)^n = (1-z)^n are va;ues of i*tan(r*PI/n) where r = 0, 1, 2, .. (n-1).
Show that roots of (1+z)^n = (1-z)^n are va;ues of i*tan(r*PI/n) where r = 0, 1, 2, .. (n-1).
Sum to n terms 1/(1.3) + 1/(3.5) + 1/(5.7) + ...
Sum to n terms 1/(1.3) + 1/(3.5) + 1/(5.7) + ...
Sum to n terms 1.(2^2) + 2.(3^2) + 3.(4^2) + ...
Sum to n terms 1.(2^2) + 2.(3^2) + 3.(4^2) + ...
Sum to n terms 1/(1.2.3) + 1/(2.3.4) + 1/(3.4.5) + ...
Sum to n terms 1/(1.2.3) + 1/(2.3.4) + 1/(3.4.5) + ...
Sum to n terms 2.3 + 3.6 + 4.11 + ...
Sum to n terms 2.3 + 3.6 + 4.11 + ...
Sum to n terms 1.3.5 + 3.5.7 + 5.7.9 + ...
Sum to n terms 1.3.5 + 3.5.7 + 5.7.9 + ...
Sum the following series to n terms and infinity: 1/9 + 2/225 + ... + n/[(2n-1)(2n+1)]^2 + ...
Sum the following series to n terms and infinity: 1/9 + 2/225 + ... + n/[(2n-1)(2n+1)]^2 + ...
Given that 1+2i is a root of x^4-3*cube(x)+8*square(x)-7x+5=0, find the other roots.
Given that 1+2i is a root of x^4-3*cube(x)+8*square(x)-7x+5=0, find the other roots.
Compute {[-1+i*sqrt(3)]/2} raised to power 6 + {[1-i*sqrt(7)]/2} raised to power 6.
Compute {[-1+i*sqrt(3)]/2} raised to power 6 + {[1-i*sqrt(7)]/2} raised to power 6.
Prove that x^5-1 = (x-1)[square(x)+2xcos(PI/5)+1][square(x)-2xcos(2*PI/5)+1].
Prove that x^5-1 = (x-1)[square(x)+2xcos(PI/5)+1][square(x)-2xcos(2*PI/5)+1].
Prove |z1+z2+z3|^2+|-z1+z2+z3|^2+|z1-z2+z3|^2+|z1+z2-z3|^2=4[|z1|^2+|z2|^2+|z3|^2].
Prove |z1+z2+z3|^2+|-z1+z2+z3|^2+|z1-z2+z3|^2+|z1+z2-z3|^2=4[|z1|^2+|z2|^2+|z3|^2].
Compute 1 + i + square(i) + cube(i) + ... + i^n for n greater than equal to 1 where i = sqrt(-1).
Compute 1 + i + square(i) + cube(i) + ... + i^n for n greater than equal to 1 where i = sqrt(-1).
If roots of square(x)+px+square(q)=0 have same absolute value then show that p divided by q is real.
If roots of square(x)+px+square(q)=0 have same absolute value then show that p divided by q is real.