The figure below is an equilateral triangle. What is the area of the triangle if the length of the sides is 6?

a_1=3 , a_n=2a^2_{n-1}-1 , find the value of \lim\limits_{n\to \infty}\dfrac{a_n}{2^na_1a_2\dots a_{n-1}}

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

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.

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.

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.

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}