Sequences and S...

If $$m * n = m+(m-1)+(m-2)+ ...... +(m-n)$$, evaluate $$7 * 5$$.

If $$a\odot b = 6\times a - 3\times b$$, evaluate $$(5\odot 3) \odot 20$$

What is the missing number in the sequence $$1, 5, 10, 16, 23, 31,$$ ____?

$$6, 10, 18, 34, 66$$
The first number in the list above is $$6$$. Determine a rule for finding each successive number in the list.

The value of the sum $$1.2.3+2.3.4+3.4.5+...$$ upto n terms =

$$m, 2m, 4m, . . . $$
The first term in the sequence above is $$m$$, and each term thereafter is equal to twice the previous term. If $$m$$ is an integer, which of the following could NOT be the sum of the first four terms of this sequence?

The terms of a sequence are defined by $$a_{n} = 3a_{n - 1} - a_{n - 2}$$ for $$n > 2$$. Find the value of $$a_{5}$$ given that $$a_{1} =4$$ and $$a_{2} = 3$$.

For all numbers a and b, let $$\displaystyle a\bigodot b$$ be defined by $$\displaystyle a\bigodot b=ab+a+b$$. Then for the numbers $$x$$, $$y$$ and $$z$$, which of the following is/are true?
I. $$\displaystyle x\bigodot y=y\bigodot x$$
II. $$\displaystyle \left( x-1 \right) \bigodot \left( x+1 \right) =\left( x\bigodot x \right) -1$$
III. $$\displaystyle x\bigodot \left( y+z \right) =\left( x\bigodot y \right) +\left( x\bigodot z \right) $$

The sum of $$1$$ st $$n$$ terms of the series
$$\cfrac { { 1 }^{ 2 } }{ 1 } +\cfrac { { 1 }^{ 2 }+{ 2 }^{ 2 } }{ 1+2 } +\cfrac { { 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 } }{ 1+2+3 } +........$$

The sum of the series,
$$\displaystyle \frac{1}{2.3}\cdot 2+ \frac{2}{3.4}\cdot 2^2+\frac{3}{4.5}\cdot 2^3+ ......$$ to n terms is _____.