Answer
On January: $ \$1,073,741,824$
Total amount during January: $ \$2,147,483,647$.
Work Step by Step
We have,
January 1: $\$1$
January 2: $\$2$
January 3: $\$4$
The pattern is Geometric sequence with ratio $r=2$.
Part 1:
We can use formula of nth term of Geometric sequence to find the amount saved on January 31.
$\therefore a_n=a_0r^{n-1}$ Where, $a_0$ is first term of sequence
Putting $a_0=1$
$r=2$
$n=31$
We get,
$a_{31}=(1)(2^{30})=\$1,073,741,824$
Part 2:
For total amount we can use summation formula for Geometric sequence, which is,
$S_n=\frac{a_0(r^n-1)}{r-1}$
Putting,
$n=31,r=2, a_0=1$
We get,
$S_{31}=\frac{(1)(2^{31}-1)}{2-1}=\$2,147,483,647$
