Jane claims that the first three terms of the sequence 1, 2, 4, ... can also be represented by an expression that is different from $T_n = \frac{1}{2} n^2 - \frac{1}{2} n + 1$. Is she correct? Justify your answer. Answer ___ $\xi =$ {Students in a college} $M =$ {Students who take Mathematics} $E =$ {Students who take Economics} $P =$ {Students who take Psychology} Represent the following using set language. There are students who take Mathematics and Economics. Answer ___ All students who take Psychology also take Mathematics. Answer ___

1. Verify that the given quadratic expression represents the first three terms.

For $T_n=\frac12n^2-\frac12n+1$:

When $n=1$, $T_1=\frac12(1)^2-\frac12(1)+1=1$.

When $n=2$, $T_2=\frac12(2)^2-\frac12(2)+1=2$.

When $n=3$, $T_3=\frac12(3)^2-\frac12(3)+1=4$.

2. Give a different expression that also produces the first three terms and conclude.

Another possible expression is

$T_n=2^{n-1}.$

Then

$T_1=2^0=1,\quad T_2=2^1=2,\quad T_3=2^2=4.$

This expression is different from $\frac12n^2-\frac12n+1$, so Jane is correct.

1. Represent students taking both Mathematics and Economics using intersection and state that the set is non-empty.

Students who take Mathematics and Economics are represented by $M\cap E$.

Since there are such students,
$M\cap E\ne \varnothing.$

1. Translate the statement into subset notation.

Since all students in $P$ are also in $M$, $P$ is a subset of $M$.

$P \subseteq M$