Substitute n=1, 2, 3, 4, 5 to find the first five sequences in the given sequence \left\{\frac{(-1)^{n-1}}{2\cdot4\cdot6\dotsm2n}\right\}

Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions.
${\displaystyle a_n=n{a}_{n-1}+a_{n-2}^2,a_0=-1,a_1=0}$

Prove that ${\displaystyle \underset{n\Rightarrow \mathrm{\infty }}{lim}\frac{n}{2n-1}=\frac{1}{2}}$

If you flip a coin 9 times, you get a sequence of Heads (H) and tails (T).
Solve each question
(a) How many different sequences of heads and tails are possible?
(b)How many different sequences of heads and tails have exactly fiveheads?
c) How many different sequences have at most 2 heads?

What is the least common multiple of 8 and 12?

Give the first six terms of this sequence
${\displaystyle c_1=4,c_2=5,\text{and}{c}_n=c_{n-1}.c_{n-2}\text{for}n\ge 3}$

There are three sequences. Which of them are arithmetic
${\displaystyle 2,4,6,8,10,..}$
${\displaystyle 2,4,8,16,32,\dots }$
${\displaystyle a,a+2,a+4,a+6,a+8,\dots }$

Which of the following sequences is NOT arithmetic?
(A) -4, 2, 8, 14, ...,
(B) 9, 4, -1, -6, ...
(C) 2, 4, 8, 16,...
(D)$\frac{1}{3},1\times \frac{1}{3},2\times \frac{1}{3},3\times \frac{1}{3}$',...