Given that Sn = 2n2 − 3n, find S6.

The formula is $S_n = 2n^2 - 3n$.
We need to find $S_6$.

The value of $S_6$ is 54.

The correct option is B.

18.

Let the number of green balls be $G = 20$.
Let the number of red balls be $R$.
The total number of balls is $T = G + R = 20 + R$.
The probability of selecting a red ball is $P(\text{red}) = \frac{\text{Number of red balls}}{\text{Total number of balls}} = \frac{R}{20+R}$.
Given that $P(\text{red}) = \frac{3}{5}$.
So, we have the equation:

Cross-multiply:

Subtract $3R$ from both sides:

Divide by 2:

There are 30 red balls.

Total balls $T = 20 + R = 20 + 30$.

There are 50 balls in the bag.

The correct option is B.

19.

The first term is $a = 20$.
The $n$-th term of an AP is given by $a_n = a + (n-1)d$, where $d$ is the common difference.
The third term is $a_3 = a + (3-1)d = a + 2d$.
The fifth term is $a_5 = a + (5-1)d = a + 4d$.

The fifth term is four-fifth of the third term:

Multiply both sides by 5 to eliminate the fraction:

Subtract $8d$ from both sides:

Subtract 100 from both sides:

Divide by 12:

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 4:

The common difference is $*-\frac{5}{3}*$.

The correct option is C.