For which value of \lambda, do the simultaneous equations 3x+4y=\lambda and 6x+8y=2 have infinite solutions. Linear Equations

Given a geometric sequence with the common ratio q (q>1) and the number of terms m, removing one of the terms from the sequence will result in a new sequence. Find the values of m and q if the new sequence is an arithmetic sequence. Arithmetic sequence

If n\in N^* such that a_1=\sqrt{2} , a_{n+1}=\sqrt{2+a_n}, show that a_n has limit and find the value of the the limit. Limit

If the sequence \{a_n\} has its first term a_1=1, a_{n+1}=\dfrac{1+4a_n+\sqrt{1+24a_n}}{16} , find the formula for general terms of the sequence. Recursive Sequence

Show that the sequence \{a_n\} is composed of terms of integers if a_0=1 and a_{n+1}=\dfrac{3a_n+\sqrt{5a^2_n-4} }{2} Recursive Sequence

If a sequence \{a_n\} has its first term a_0=1, a_{n+1}=\dfrac{7a_n+\sqrt{45{a_n}^2-36}}{2}, (n\in N), show that

If a, b, c are three sides of the \triangle ABC and A, B, C are their corresponding vertex, show that a^2\tan\dfrac{A}{2}+b^2\tan\dfrac{B} {2}+c^2\tan\dfrac{C}{2}\geq 4S in which S is the area of the triangle. Trigonometry

Compute the smallest positive integer n for which\sqrt{100+\sqrt{n} } +\sqrt{100-\sqrt{n} } is an integer Cube root