Question

You are paying for your niece's education every month; contribution decreasing monthly at a constant rate. If your 20th contribution (after 19 months) was $483 and 43rd contribution was $345, what was your first contribution?

Answer 1

Answer:

The first contribution was 637.77

Step-by-step explanation:

Geometric Sequences

It a type sequence in which each term is computed as the previous term by a constant number. The general expression for a geometric sequence is

$\displaystyle a_n=a_1.r^{n-1},\ n>0$

If we know two terms of the sequence, say n=k and n=p, then

$\displaystyle a_k=a_1.r^{k-1}$

and

$\displaystyle a_p=a_1.r^{p-1}$

We can determine the values of $a_1$ and r, by manipulating both equations

We know that

$a_{20}=483,\ a_{43}=345,\ so$

$\displaystyle a_{20}=483=a_1.r^{20-1}$

$\displaystyle a_{43}=345=a_1.r^{43-1}$

Dividing both expressions, we have

$\displaystyle \frac{a_{43}}{a_{20}}=\frac{r^{42}}{r^{19}}$

Solving for r

$\displaystyle r^{23}=\frac{345}{483}$

$\displaystyle r=\sqrt[23]{\frac{345}{483}}$

$\displaystyle r=0.9855$

Now we use

$\displaystyle a_{20}=483=a_1.r^{20-1}$

to compute $a_1$

$\displaystyle a_1=\frac{a_{20}}{r_{19}}=\frac{483}{0.9855^{19}}$

$\boxed{a_1=637.77}$