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

The first contribution was 637.77

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 (a_(43))/(a_(20))=(r^(42))/(r^(19))`

Solving for r

`\displaystyle r^(23)=(345)/(483)`

`\displaystyle r=\sqrt[23]{(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=(a_(20))/(r_(19))=(483)/(0.9855^(19))`

`\boxed{a_1=637.77}`